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KataHex

2026-04-18T12:47:38Z

Hexanna: mention the 20250131 net; fix links

'''KataHex''' is a free and open-source computer Hex program, capable of defeating top-level human players. It implements Monte Carlo tree search with a convolutional neural network providing position evaluation and policy guidance.

== History and versions ==

KataHex is based on [https://en.wikipedia.org/wiki/KataGo KataGo], a computer Go program developed by David Wu that was first released on 27 February 2019. It was adapted for Hex by "HZY" between February 2020 and May 2022. While initially unnamed, the Hex-adaptation of KataGo quickly became known as "KataHex" among Hex players.

The HZY implementation of KataHex speaks a non-standard dialect of [[GTP]] and can only interact with a specially modified version of the Go GUI known as LizzieYzy. A further adaptation of KataHex that is capable of interfacing with [[Hexgui]] was made by [[User:Selinger|Selinger]].

== Pre-trained networks ==

HZY initially trained the neural network model (20220618) on two NVIDIA GeForce RTX 2080 Ti GPUs for about 20 days on 13x13. They then up-trained the network on 19x19 for one day, and on 27x27 for an additional day. [https://zhuanlan-zhihu-com.translate.goog/p/476464087?_x_tr_sl=auto&_x_tr_tl=en According to HZY], the up-trained 19x19 network is relatively reliable, but the 27x27-network is not.

A newer model, 20240812, is trained on two RTX 4090 GPUs for 2 months on 15x15, 15 days on 19x19, and 3 days on 27x27. This model uses a b18c384nbt architecture (18 block, 384 channel, [https://github.com/lightvector/KataGo/blob/master/docs/KataGoMethods.md#nested-bottleneck-residual-nets nested bottleneck residual net]).

The newest model, 20250131, uses a b28c512nbt architecture. At equal node counts, it is around 100–200 Elo stronger than 20240812. However, because it is a larger network, each evaluation is around 2.8× more costly. (Evaluation time for a network with B blocks and C channels is typically proportional to B × C<sup>2</sup>.) Under equal time controls, the two networks are much closer in strength.

Running KataHex requires both a neural net model (the "weights"), and an engine to load the weights. Each neural net is able to play Hex not only at the size it was trained on, but also smaller and larger sizes, as long as the engine is compiled to support the size. (However, the net will not play particularly well on sizes larger than it was trained on.)

"KataHex" refers generally to the KataHex program, but often more specifically to the pre-trained 19x19 neural network made by HZY.

== Limitations ==

Since KataHex was only trained on self-play, it does not always do well when asked to play from an arbitrary starting position. Older nets are particularly thrown off if the starting position does not have the same number of black and white stones, but the issue seems to have been mitigated with the 20240812 net.

== Links ==

'''Program:'''

* David Wu's original KataGo: https://github.com/lightvector/KataGo
* HZY's KataHex: https://github.com/hzyhhzy/KataGo/tree/Hex2024
* Selinger's modifications: https://github.com/selinger/katahex/tree/Hex2024

'''Pre-trained network models:'''

* 20250131 (latest): https://github.com/hzyhhzy/KataGomo/releases/Hex_20250131
** (weights only): [https://github.com/hzyhhzy/KataGomo/releases/download/Hex_20250131/hex3_27x_b28.bin.gz hex3_27x_b28.bin.gz]
** (supports move limit mode, but slightly weaker): [https://github.com/hzyhhzy/KataGomo/releases/download/Hex_20250131/hex3_mm19x_b28.bin.gz hex3_mm19x_b28.bin.gz]
* 20240812: https://github.com/hzyhhzy/KataGomo_fork/releases/Hex_20240812
** (weights only): [https://drive.usercontent.google.com/download?id=1YeqRvAYs7YjtPh0xBbDnHxeo2xrLEOdX hex27x3.bin.gz]
* 20220618: [https://drive.google.com/file/d/1xMvP_75xgo0271nQbmlAJ40rvpKiFTgP/view katahex_model_20220618.bin.gz]

'''GUIs:'''

* The 20240812 net is bundled with a modified LizzieYzy GUI: https://github.com/hzyhhzy/KataGo/releases/tag/Hex_20240812
* An older net with the modified LizzieYzy GUI: [https://drive.google.com/file/d/1qbTTmPFiUkM_346DeKS1E9gJR-roNH63/view KataHex_LizzieYZY_20220313.zip]
* Selinger's KataHex works with HexGUI: https://github.com/selinger/hexgui

== See also ==

* [[Strategic advice from KataHex]]

[[Category:Computer Hex]]
[[category:Hex playing program]]

User:Hexanna

2026-04-08T02:43:28Z

Hexanna: tweak stone fraction claims to match the local pattern explorer

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata] is a GUI and CLI for analyzing Hex that I vibe-coded. It's still under active development, but I already use it as my main GUI.
* One of the GUI's main advantages (and a primary reason I created it) is that it's fast and responsive. It's not as useful for "traditional" analysis as some other GUI's — it doesn't support adding comments to moves, for example.
* The CLI is primarily intended for AI coding tools and scripts. For example, I have used it to proofread and validate some of the KataHex eval claims made in my articles.

[https://hexanna1.github.io/hex-study/ Hex Study Tools] is a collection of interactive databases:
* [https://hexanna1.github.io/hex-study/patterns.html Local pattern explorer] with up to 5-stone local patterns
* [https://hexanna1.github.io/hex-study/joseki.html Joseki explorer] with acute and obtuse corner sequences
* [https://hexanna1.github.io/hex-study/openings.html Opening explorer] with opening lines on sizes 11–14 and 17
* Of these tools, I put the most thought into the design of the local pattern explorer, and it's probably the most useful one for improving at Hex.

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.4-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.35-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one where Red plays (c), so this is at least a 0.35-stone mistake. KataHex says Blue can do even better responding with tenuki or (e), so it thinks (d) is actually a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex says it's a 0.2-stone mistake.
:** (f): This is a retroactive near allow-cut. Blue can play at (a) or (b) to reach a position congruent to the one where Red plays (e), so it's at least a 0.2-stone mistake. Blue tenuki may be slightly stronger; KataHex says (f) is a 0.25-stone mistake because of that.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* Deviations between KataHex and our mental model are unsurprising and don't invalidate the mental model — our reasoning was a bit handwavy, Blue often has stronger alternatives, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.2-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.25-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.35-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.4-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.4-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex nearly agrees and says it's a 0.35-stone mistake.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.5-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

===Miscellaneous topics===
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from [[KataHex]]==

* Swap map for 19×19 generated with KataHex net 20240812 using [https://github.com/hexanna1/hexata Hexata], with around 30k–40k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 84:(d3 p17)
19:(e3 o17)
108:(f3 n17)
118:(g3 m17)
107:(h3 l17)
127:(i3 k17)
140:(j3 j17)
136:(k3 i17)
140:(l3 h17)
72:(m3 g17)
13:(n3 f17)
73:(o3 e17)
70:(p3 d17)
142:(h4 l16)
86:(i4 k16)
76:(j4 j16)
92:(k4 i16)
104:(l4 h16)
165:(m4 g16)
158:(i5 k15)
124:(j5 j15)
161:(k5 i15)
127:(q2 c18)
214:(p2 d18)
136:(b17 r3)
250:(b18 r2)
121:(a2 s18)
118:(b2 r18)
89:(c2 q18)
220:(d2 p18)
127:(a3 s17)
184:(b3 r17)
199:(c3 q17)
126:(a4 s16)
115:(b4 r16)
204:(a5 s15)
146:(a6 s14)
136:(a7 s13)
195:(a8 s12)
167:(a9 s11)
87:(a10 s10)
126:(a11 s9)
168:(a12 s8)
111:(a13 s7)
92:(a14 s6)
53:(a15 s5)
73:(a16 s4)
164:(a17 s3)
242:(a18 s2)
70:(a19 s1)
523:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2026-03-29T18:52:37Z

Hexanna:

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata] is a GUI and CLI for analyzing Hex that I vibe-coded. It's still under active development, but I already use it as my main GUI.
* One of the GUI's main advantages (and a primary reason I created it) is that it's fast and responsive. It's not as useful for "traditional" analysis as some other GUI's — it doesn't support adding comments to moves, for example.
* The CLI is primarily intended for AI coding tools and scripts. For example, I have used it to proofread and validate some of the KataHex eval claims made in my articles.

[https://hexanna1.github.io/hex-study/ Hex Study Tools] is a collection of interactive databases:
* [https://hexanna1.github.io/hex-study/patterns.html Local pattern explorer] with up to 5-stone local patterns
* [https://hexanna1.github.io/hex-study/joseki.html Joseki explorer] with acute and obtuse corner sequences
* [https://hexanna1.github.io/hex-study/openings.html Opening explorer] with opening lines on sizes 11–14 and 17
* Of these tools, I put the most thought into the design of the local pattern explorer, and it's probably the most useful one for improving at Hex.

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

===Miscellaneous topics===
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from [[KataHex]]==

* Swap map for 19×19 generated with KataHex net 20240812 using [https://github.com/hexanna1/hexata Hexata], with around 30k–40k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 84:(d3 p17)
19:(e3 o17)
108:(f3 n17)
118:(g3 m17)
107:(h3 l17)
127:(i3 k17)
140:(j3 j17)
136:(k3 i17)
140:(l3 h17)
72:(m3 g17)
13:(n3 f17)
73:(o3 e17)
70:(p3 d17)
142:(h4 l16)
86:(i4 k16)
76:(j4 j16)
92:(k4 i16)
104:(l4 h16)
165:(m4 g16)
158:(i5 k15)
124:(j5 j15)
161:(k5 i15)
127:(q2 c18)
214:(p2 d18)
136:(b17 r3)
250:(b18 r2)
121:(a2 s18)
118:(b2 r18)
89:(c2 q18)
220:(d2 p18)
127:(a3 s17)
184:(b3 r17)
199:(c3 q17)
126:(a4 s16)
115:(b4 r16)
204:(a5 s15)
146:(a6 s14)
136:(a7 s13)
195:(a8 s12)
167:(a9 s11)
87:(a10 s10)
126:(a11 s9)
168:(a12 s8)
111:(a13 s7)
92:(a14 s6)
53:(a15 s5)
73:(a16 s4)
164:(a17 s3)
242:(a18 s2)
70:(a19 s1)
523:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2026-03-29T18:51:25Z

Hexanna: remove old stuff

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata] is a GUI and CLI for analyzing Hex that I vibe-coded. It's still under active development, but I already use it as my main GUI.
* One of the GUI's main advantages (and a primary reason I created it) is that it's fast and responsive. It's not as useful for "traditional" analysis as some other GUI's — it doesn't support adding comments to moves, for example.
* The CLI is primarily intended for AI coding tools and scripts. For example, I have used it to proofread and validate some of the KataHex eval claims made in my articles.

[https://hexanna1.github.io/hex-study/ Hex Study Tools] is a collection of interactive databases:
* [https://hexanna1.github.io/hex-study/patterns.html Local pattern explorer] with up to 5-stone local patterns
* [https://hexanna1.github.io/hex-study/joseki.html Joseki explorer] with acute and obtuse corner sequences
* [https://hexanna1.github.io/hex-study/openings.html Opening explorer] with opening lines on sizes 11–14 and 17
* Of these tools, I put the most thought into the design of the local pattern explorer, and it's probably the most useful one for improving at Hex.

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from [[KataHex]]==

* Swap map for 19×19 generated with KataHex net 20240812 using [https://github.com/hexanna1/hexata Hexata], with around 30k–40k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 84:(d3 p17)
19:(e3 o17)
108:(f3 n17)
118:(g3 m17)
107:(h3 l17)
127:(i3 k17)
140:(j3 j17)
136:(k3 i17)
140:(l3 h17)
72:(m3 g17)
13:(n3 f17)
73:(o3 e17)
70:(p3 d17)
142:(h4 l16)
86:(i4 k16)
76:(j4 j16)
92:(k4 i16)
104:(l4 h16)
165:(m4 g16)
158:(i5 k15)
124:(j5 j15)
161:(k5 i15)
127:(q2 c18)
214:(p2 d18)
136:(b17 r3)
250:(b18 r2)
121:(a2 s18)
118:(b2 r18)
89:(c2 q18)
220:(d2 p18)
127:(a3 s17)
184:(b3 r17)
199:(c3 q17)
126:(a4 s16)
115:(b4 r16)
204:(a5 s15)
146:(a6 s14)
136:(a7 s13)
195:(a8 s12)
167:(a9 s11)
87:(a10 s10)
126:(a11 s9)
168:(a12 s8)
111:(a13 s7)
92:(a14 s6)
53:(a15 s5)
73:(a16 s4)
164:(a17 s3)
242:(a18 s2)
70:(a19 s1)
523:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2026-03-29T18:42:36Z

Hexanna: opening database

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata] is a GUI and CLI for analyzing Hex that I vibe-coded. It's still under active development, but I already use it as my main GUI.
* One of the GUI's main advantages (and a primary reason I created it) is that it's fast and responsive. It's not as useful for "traditional" analysis as some other GUI's — it doesn't support adding comments to moves, for example.
* The CLI is primarily intended for AI coding tools and scripts. For example, I have used it to proofread and validate some of the KataHex eval claims made in my articles.

[https://hexanna1.github.io/hex-study/ Hex Study Tools] is a collection of interactive databases:
* [https://hexanna1.github.io/hex-study/patterns.html Local pattern explorer] with up to 5-stone local patterns
* [https://hexanna1.github.io/hex-study/joseki.html Joseki explorer] with acute and obtuse corner sequences
* [https://hexanna1.github.io/hex-study/openings.html Opening explorer] with opening lines on sizes 11–14 and 17
* Of these tools, I put the most thought into the design of the local pattern explorer, and it's probably the most useful one for improving at Hex.

==current projects==

* working on Hexata GUI and CLI
* using Hexata CLI in vibe-coded scripts to understand Hex better
** this is harder than I anticipated; there are a lot of decisions to make and a lot of things that could go wrong
** for example: I've been trying to understand the strength of stones relative to local patterns in a more systematic way (removing bias introduced by the board edges and other sources of noise). I have always preferred using larger boards like 25×25 because the center of the board is far away from the edges, and katahex evals are further away from unstable 0% or 100% regions. I checked whether my methodology gave a consistent result across board sizes. unfortunately, it didn't, even though "the strength of stones relative to local patterns" is a concept that's theoretically independent of board size. preliminary investigation of this inconsistency shows that — despite the application of appropriate noise-reduction methods — there appears to be a systematic bias to its evals (on katahex 20240812) that becomes visible around the size 22→23 boundary and gets progressively worse as board size increases, as illustrated in [https://github.com/hexanna1/hex-study/blob/main/results/board_size_vs_canonical_key_stone_fraction_d3.png this graph]. so, many of my 25×25 analyses on local patterns are slightly wrong (luckily, I didn't use a larger size like 31 where the bias is more severe); a smaller size like 20–22 may be optimal for this net. (usual caveat: this finding is specific to katahex 20240812 and does not generalize to other nets. it does not imply that size 25 is categorically worse than sizes 20–22 for other kinds of analyses.)

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from [[KataHex]]==

* Swap map for 19×19 generated with KataHex net 20240812 using [https://github.com/hexanna1/hexata Hexata], with around 30k–40k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 84:(d3 p17)
19:(e3 o17)
108:(f3 n17)
118:(g3 m17)
107:(h3 l17)
127:(i3 k17)
140:(j3 j17)
136:(k3 i17)
140:(l3 h17)
72:(m3 g17)
13:(n3 f17)
73:(o3 e17)
70:(p3 d17)
142:(h4 l16)
86:(i4 k16)
76:(j4 j16)
92:(k4 i16)
104:(l4 h16)
165:(m4 g16)
158:(i5 k15)
124:(j5 j15)
161:(k5 i15)
127:(q2 c18)
214:(p2 d18)
136:(b17 r3)
250:(b18 r2)
121:(a2 s18)
118:(b2 r18)
89:(c2 q18)
220:(d2 p18)
127:(a3 s17)
184:(b3 r17)
199:(c3 q17)
126:(a4 s16)
115:(b4 r16)
204:(a5 s15)
146:(a6 s14)
136:(a7 s13)
195:(a8 s12)
167:(a9 s11)
87:(a10 s10)
126:(a11 s9)
168:(a12 s8)
111:(a13 s7)
92:(a14 s6)
53:(a15 s5)
73:(a16 s4)
164:(a17 s3)
242:(a18 s2)
70:(a19 s1)
523:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2026-03-24T02:30:28Z

Hexanna: joseki explorer

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata] is a GUI and CLI for analyzing Hex that I vibe-coded. It's still under active development, but I already use it as my main GUI.
* One of the GUI's main advantages (and a primary reason I created it) is that it's fast and responsive. It's not as useful for "traditional" analysis as some other GUI's — it doesn't support adding comments to moves, for example.
* The CLI is primarily intended for AI coding tools and scripts. For example, I have used it to proofread and validate some of the KataHex eval claims made in my articles.

[https://hexanna1.github.io/hex-study/ Hex Study Tools] is a collection of interactive tools/mini-databases. There are pages for [https://hexanna1.github.io/hex-study/patterns.html local patterns] and [https://hexanna1.github.io/hex-study/joseki.html josekis].
* The local pattern explorer is the more refined tool, and probably the more useful one for improving at Hex.

==current projects==

* working on Hexata GUI and CLI
* using Hexata CLI in vibe-coded scripts to understand Hex better
** this is harder than I anticipated; there are a lot of decisions to make and a lot of things that could go wrong
** for example: I've been trying to understand the strength of stones relative to local patterns in a more systematic way (removing bias introduced by the board edges and other sources of noise). I have always preferred using larger boards like 25×25 because the center of the board is far away from the edges, and katahex evals are further away from unstable 0% or 100% regions. I checked whether my methodology gave a consistent result across board sizes. unfortunately, it didn't, even though "the strength of stones relative to local patterns" is a concept that's theoretically independent of board size. preliminary investigation of this inconsistency shows that — despite the application of appropriate noise-reduction methods — there appears to be a systematic bias to its evals (on katahex 20240812) that becomes visible around the size 22→23 boundary and gets progressively worse as board size increases, as illustrated in [https://github.com/hexanna1/hex-study/blob/main/results/board_size_vs_canonical_key_stone_fraction_d3.png this graph]. so, many of my 25×25 analyses on local patterns are slightly wrong (luckily, I didn't use a larger size like 31 where the bias is more severe); a smaller size like 20–22 may be optimal for this net. (usual caveat: this finding is specific to katahex 20240812 and does not generalize to other nets. it does not imply that size 25 is categorically worse than sizes 20–22 for other kinds of analyses.)

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from [[KataHex]]==

* Swap map for 19×19 generated with KataHex net 20240812 using [https://github.com/hexanna1/hexata Hexata], with around 30k–40k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 84:(d3 p17)
19:(e3 o17)
108:(f3 n17)
118:(g3 m17)
107:(h3 l17)
127:(i3 k17)
140:(j3 j17)
136:(k3 i17)
140:(l3 h17)
72:(m3 g17)
13:(n3 f17)
73:(o3 e17)
70:(p3 d17)
142:(h4 l16)
86:(i4 k16)
76:(j4 j16)
92:(k4 i16)
104:(l4 h16)
165:(m4 g16)
158:(i5 k15)
124:(j5 j15)
161:(k5 i15)
127:(q2 c18)
214:(p2 d18)
136:(b17 r3)
250:(b18 r2)
121:(a2 s18)
118:(b2 r18)
89:(c2 q18)
220:(d2 p18)
127:(a3 s17)
184:(b3 r17)
199:(c3 q17)
126:(a4 s16)
115:(b4 r16)
204:(a5 s15)
146:(a6 s14)
136:(a7 s13)
195:(a8 s12)
167:(a9 s11)
87:(a10 s10)
126:(a11 s9)
168:(a12 s8)
111:(a13 s7)
92:(a14 s6)
53:(a15 s5)
73:(a16 s4)
164:(a17 s3)
242:(a18 s2)
70:(a19 s1)
523:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2026-03-18T00:11:33Z

Hexanna: hex local pattern explorer

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata] is a GUI and CLI for analyzing Hex that I vibe-coded. It's still under active development, but I already use it as my main GUI.
* One of the GUI's main advantages (and a primary reason I created it) is that it's fast and responsive. It's not as useful for "traditional" analysis as some other GUI's — it doesn't support adding comments to moves, for example.
* The CLI is primarily intended for AI coding tools and scripts. For example, I have used it to proofread and validate some of the KataHex eval claims made in my articles.

[https://hexanna1.github.io/hex-study/ Hex Local Pattern Explorer]

==current projects==

* working on Hexata GUI and CLI
* using Hexata CLI in vibe-coded scripts to understand Hex better
** this is harder than I anticipated; there are a lot of decisions to make and a lot of things that could go wrong
** for example: I've been trying to understand the strength of stones relative to local patterns in a more systematic way (removing bias introduced by the board edges and other sources of noise). I have always preferred using larger boards like 25×25 because the center of the board is far away from the edges, and katahex evals are further away from unstable 0% or 100% regions. I checked whether my methodology gave a consistent result across board sizes. unfortunately, it didn't, even though "the strength of stones relative to local patterns" is a concept that's theoretically independent of board size. preliminary investigation of this inconsistency shows that — despite the application of appropriate noise-reduction methods — there appears to be a systematic bias to its evals (on katahex 20240812) that becomes visible around the size 22→23 boundary and gets progressively worse as board size increases, as illustrated in [https://github.com/hexanna1/hex-study/blob/main/results/board_size_vs_canonical_key_stone_fraction_d3.png this graph]. so, many of my 25×25 analyses on local patterns are slightly wrong (luckily, I didn't use a larger size like 31 where the bias is more severe); a smaller size like 20–22 may be optimal for this net. (usual caveat: this finding is specific to katahex 20240812 and does not generalize to other nets. it does not imply that size 25 is categorically worse than sizes 20–22 for other kinds of analyses.)

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from [[KataHex]]==

* Swap map for 19×19 generated with KataHex net 20240812 using [https://github.com/hexanna1/hexata Hexata], with around 30k–40k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 84:(d3 p17)
19:(e3 o17)
108:(f3 n17)
118:(g3 m17)
107:(h3 l17)
127:(i3 k17)
140:(j3 j17)
136:(k3 i17)
140:(l3 h17)
72:(m3 g17)
13:(n3 f17)
73:(o3 e17)
70:(p3 d17)
142:(h4 l16)
86:(i4 k16)
76:(j4 j16)
92:(k4 i16)
104:(l4 h16)
165:(m4 g16)
158:(i5 k15)
124:(j5 j15)
161:(k5 i15)
127:(q2 c18)
214:(p2 d18)
136:(b17 r3)
250:(b18 r2)
121:(a2 s18)
118:(b2 r18)
89:(c2 q18)
220:(d2 p18)
127:(a3 s17)
184:(b3 r17)
199:(c3 q17)
126:(a4 s16)
115:(b4 r16)
204:(a5 s15)
146:(a6 s14)
136:(a7 s13)
195:(a8 s12)
167:(a9 s11)
87:(a10 s10)
126:(a11 s9)
168:(a12 s8)
111:(a13 s7)
92:(a14 s6)
53:(a15 s5)
73:(a16 s4)
164:(a17 s3)
242:(a18 s2)
70:(a19 s1)
523:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

Computer Hex

2026-03-13T22:45:27Z

Hexanna: updated Hexata description

This page lists some software programs and programming topics that may be of interest to [[Hex]] players. The programs include AI opponents and tools for analysing completed games.

More complete or up-to-date information is welcome.

== AI techniques used in Hex ==

*[[Minimax (computer)|Minimax]] and alpha-beta search were used by [[Queenbee]].
*[[UCT]] is used in MoHex.

== Programs with AI ==

There are several computer programs which play Hex.

=== Available programs ===

{| class="wikitable"
|-
! Program !! Platforms !! Remarks
|-
| [[KataHex]] || Linux, Windows || By HZY. As of 2024, the strongest available Hex program. Very strong AI, open source.
|-
| [[MoHex]] || Linux || Was the strongest available Hex program in 2010. It uses the UCT-Monte Carlo approach and is developed at the University of Alberta by Philip Henderson, Broderick Arneson and Ryan Hayward. Also has an efficient solver.
|-
| [[Hexy]] || Windows || The second strongest program available. It was the first program to use virtual connections and was champion of the 5th Computer Olympiad in 2000.
|-
| [[Wolve]] || Linux || Gold medallist of 2008 Computer Olympiads.
|-
| [[Six]] || Linux, Unix, Windows || by Gábor Melis.
|-
| [http://www.mattesmedjan.se/hexilla/ Hexilla] || Java || By Jonatan Rydh, released in October 2009.
|-
| [https://play.google.com/store/apps/details?id=com.game.hex Hex] || Android || By Five Factorial, released in January 2017. It uses MoHex engine on Expert level.
|}

=== Mac platform ===

No known programs for the Mac are available. There is a work around by using an emulator such as BlueStacks that allows Android programs to be run.

=== Unavailable programs ===

{| class="wikitable"
|-
! Program !! Platforms !! Remarks
|-
| [[Mongoose]] || || By [[Yngvi Björnsson]], [[Ryan Hayward]], Mike Johanson, Morgan Kan, and Nathan Po.
|-
| [[Queenbee]] || || By [[Jack van Rijswijck]]. Won silver at the London 2000 CGO.
|-
| [[Hexy (iPhone)|Hexy]] || iPhone || Despite using the same name, this program has no relation to [[Hexy]]. It was released in November 2008, offers an AI opponent; the AI appears to be a custom design and hasn't been rated. As of 26 December 2019 it is no longer available.
|-
| [https://itunes.apple.com/app/id423845369 Hexatious] || iPad, iPhone || Released in August 2009, appears to offer a stronger AI than the iPhone Hexy app (in particular, Hexatious easily beats the other iPhone app in head-to-head competition). As of 26 December 2019 it is no longer available.
|-
| [https://itunes.apple.com/app/id397349481 Hex Nash] || iPad, iPhone || Released February 2011, no AI but supports online asynchronous play and local play. As of 26 December 2019 it is no longer available.
|}

== Non playing programs ==

=== Front End ===
* [[HexGui]] is a graphical user interface designed by [[Broderick Arneson]] ("ab"). It can be used as an interactive game board to try out plays and variations, and it can also be used as a front end for any computer Hex program that can communicate via [[GTP]]. It works well as a front-end to [[MoHex]]. HexGui can read and write the [[Smart Game Format]]. An up-to-date version of HexGui is available from [https://github.com/selinger/hexgui GitHub].
* [https://github.com/hexanna1/hexata Hexata] is a lightweight, keyboard-first GUI designed by [[User:Hexanna|Hexanna]]. It is written in Python and can interact with [[KataHex]]. Hexata can import/export to HexWorld links.

=== Reviewing and Editing Programs ===

* [https://minortriad.com/ahex.html AHex] by [[User:Tom239|Tom Ace]] lets you analyze Hex and Havannah games and can import games from littlegolem.net.
* [http://canyon23.net/jgame/README_hex.html JHex] by Kevin lets you analyse a game, and databases of games.
* [http://www.drking.org.uk/hexagons/hex KHex] by David King is a tool for reviewing games. Very well suited for sharing commented games (it exports games in [[Smart Game Format]]).
* [http://www.drking.org.uk/hexagons/hex KHex18] by David King is an online app for reviewing games, which can read LittleGolem game text.

== Protocols ==

* [[GTP]] is a text-based protocol for interacting with Hex software. It is based on the Go Text Protocol, and allows Hex software to interact with Hex strategy engines.

== File formats ==

* The [[Smart Game Format]] (SGF) is a file format for storing annotated game trees. The format nor only stores a sequence of moves comprising a game, but can also contain variations (several different games played out from the same position), as well as comments on every move in the game.

== External link ==

=== Articles ===

*Anshelevich, Vadim V. [http://home.earthlink.net/~vanshel/VAnshelevich-ARTINT.pdf A hierarchical approach to computer Hex].
*van Rijswijck, Jack. [http://home.fuse.net/swmeyers/y-hex.pdf Search and evaluation in Hex].
*Rasmussen, Rune K. and Maire, Frederic D. and Hayward, Ross F. (2006) [http://eprints.qut.edu.au/5121/1/5121_1.pdf A Move Generating Algorithm for Hex Solvers].
*Rasmussen, Rune K. (2008) [http://eprints.qut.edu.au/18616/1/01Thesis.pdf Algorithmic approaches for playing and solving Shannon games] (PhD Thesis).

== See also ==
[[History of computer Hex]]

The [[ICGA|International Computer Games Association]] also has some [http://www.cs.unimaas.nl/icga/games/hex/ information on Hex]. They organize an annual [[Computer Olympiad]], which also covers Hex.

[[category:Computer Hex]]

User:Hexanna

2026-03-13T22:35:21Z

Hexanna: Hexata has branching move history now

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata] is a GUI and CLI for analyzing Hex that I vibe-coded. It's still under active development, but I already use it as my main GUI.
* One of the GUI's main advantages (and a primary reason I created it) is that it's fast and responsive. It's not as useful for "traditional" analysis as some other GUI's — it doesn't support adding comments to moves, for example.
* The CLI is primarily intended for AI coding tools and scripts. For example, I have used it to proofread and validate some of the KataHex eval claims made in my articles.

==current projects==

* working on Hexata GUI and CLI
* using Hexata CLI in vibe-coded scripts to understand Hex better
** this is harder than I anticipated; there are a lot of decisions to make and a lot of things that could go wrong
** for example: I've been trying to understand the strength of stones relative to local patterns in a more systematic way (removing bias introduced by the board edges and other sources of noise). I have always preferred using larger boards like 25×25 because the center of the board is far away from the edges, and katahex evals are further away from unstable 0% or 100% regions. I checked whether my methodology gave a consistent result across board sizes. unfortunately, it didn't, even though "the strength of stones relative to local patterns" is a concept that's theoretically independent of board size. preliminary investigation of this inconsistency shows that — despite the application of appropriate noise-reduction methods — there appears to be a systematic bias to its evals (on katahex 20240812) that becomes visible around the size 22→23 boundary and gets progressively worse as board size increases. so, many of my 25×25 analyses on local patterns are slightly wrong (luckily, I didn't use a larger size like 31 where the bias is more severe); a smaller size like 20–22 may be optimal for this net. (usual caveat: this finding is specific to katahex 20240812 and does not generalize to other nets. it does not imply that size 25 is categorically worse than sizes 20–22 for other kinds of analyses.)

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from [[KataHex]]==

* Swap map for 19×19 generated with KataHex net 20240812 using [https://github.com/hexanna1/hexata Hexata], with around 30k–40k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 84:(d3 p17)
19:(e3 o17)
108:(f3 n17)
118:(g3 m17)
107:(h3 l17)
127:(i3 k17)
140:(j3 j17)
136:(k3 i17)
140:(l3 h17)
72:(m3 g17)
13:(n3 f17)
73:(o3 e17)
70:(p3 d17)
142:(h4 l16)
86:(i4 k16)
76:(j4 j16)
92:(k4 i16)
104:(l4 h16)
165:(m4 g16)
158:(i5 k15)
124:(j5 j15)
161:(k5 i15)
127:(q2 c18)
214:(p2 d18)
136:(b17 r3)
250:(b18 r2)
121:(a2 s18)
118:(b2 r18)
89:(c2 q18)
220:(d2 p18)
127:(a3 s17)
184:(b3 r17)
199:(c3 q17)
126:(a4 s16)
115:(b4 r16)
204:(a5 s15)
146:(a6 s14)
136:(a7 s13)
195:(a8 s12)
167:(a9 s11)
87:(a10 s10)
126:(a11 s9)
168:(a12 s8)
111:(a13 s7)
92:(a14 s6)
53:(a15 s5)
73:(a16 s4)
164:(a17 s3)
242:(a18 s2)
70:(a19 s1)
523:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

Foiling

2026-03-12T23:21:15Z

Hexanna: update link

To '''foil''' a [[ladder escape]] means to make a move which prevents an [[outpost]] from being used as a [[ladder escape]], and also [[Intrusion|intrudes]] on the outpost's [[connection]] to the [[edge]].

== Example ==

Consider the following position:

<hex>C8 R8 Q1 Vc6 Hb8 Vf6 He6 Vf3 Vd4 Hd6 Hc1</hex>

[[Red (player)|Red]] has just played f6. In his next move he can either start a [[ladder]] at c7, using f6 as a ladder escape, or he can play g4, making an unbreakable connection from top to bottom. Thus f6 [[double threat|threatens two different connections]].

However it does not secure Red a connection, because there is one vulnerable cell, namely e7:

<hex>C8 R8 Q1 Vc6 Hb8 Vf6 He6 Vf3 Vd4 Hd6 Hc1 He7</hex>

If [[Blue (player)|Blue]] plays here, he prevents the use of f6 as a ladder escape, and he also intrudes on its [[edge template]] to the bottom. In fact in this position Blue [[win]]s.

So to foil a ladder escape you make a move on the row below the outpost, in the direction of where the ladder will be coming from. Are there other ways to foil?

== Foiling does not always work ==

Consider the following position, which is almost equal the one in the first diagram:

<hex>C8 R8 Q1 Vc6 Hb8 Vf6 He5 Vf3 Vd4 Hd6 Hc1</hex>

If Blue tries to foil f6 now, Red responds at f7:

<hex>C8 R8 Q1 Vc6 Hb8 Vf6 He5 Vf3 Vd4 Hd6 Hc1 He7 Vf7</hex>

Observe that the ladder still works, and so does the connection via g4. Since Blue only can stop one of these two, Red wins.

== When does foiling work? ==

In general, it is difficult to figure out when a ladder escape can be foiled. There are some simple rules that apply in some cases.

* A ladder escape fork on the second row is unfoilable.

* A ladder escape fork on the third row is unfoilable if the cell marked "*" is empty, and is not required for the "connection up":

<hex>R5 C10 Vh3 Sg3 Vb1 Vb2 Vb3 Ha5 Hc3 Hc2 Hc1 Pg4</hex>

However, if the cell marked "*" is occupied by Blue, the ladder escape fork can often be foiled; in that case, playing at "+" is the only way of foiling it. Also, if the cell marked "*" is empty, but is required for Red's threatened upward connection, the fork may be foilable by playing at "+", as in the following example:

<hex>R5 C10 Vh3 Sg3 Vb1 Vb2 Vb3 Ha5 Hc3 Hc2 Hc1 Pg4 Hd3 He3 Hf3 V10f1 Hh1 Hi1 Hi2</hex>

Because the cell marked "*" is required for Red's threatened connection to 10, the ladder escape fork is foilable by playing at "+" (but not by playing at "*").

* A ladder escape fork on the fourth row is more complicated. If a 2nd row ladder is already approaching, the fork is unfoilable if the cells marked "*" both are empty (and not required for the "connection up"). Otherwise, it may be foilable, and in that case, playing in one of the cells marked "+" is the only way to foil it.

<hex>R5 C10 Vh2 Se3 Sg2 Vb1 Vb2 Vb3 Hc3 Ha5 Hc2 Hc1 Pe4 Pg3</hex>

If the approaching ladder is a 3rd row ladder, the fork is typically foilable by playing at "+".

<hex>R5 C10 Vh2 Vb1 Vb2 Vb3 Hb4 Hc2 Hc1 Hd2 Ph3</hex>

The foil may not work if Red has a lot of space. For example, the following position is winning for Red (with Blue to move, and assuming "*" connects to the top edge), but Red needs at least the amount of space shown. If any one of the empty cells is occupied by Blue, the position is foilable.

<hex>R6 C9 Vg3 Va2 Va3 Va4 Ha5 Hb3 Hb2 Hc2 Hd2 Va1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Sh1 Hi1</hex>

To practice: [https://hexworld.org/board/#9x6c1,a1b1a2b2a3b3a4a5g3c1:pc2:pd1:pd2:pe1:pf1:pg1:pi1:p HexWorld link]

== Fishing move ==

Playing a foilable move in the hope that the opponent doesn't know how to foil is sometimes called a ''fishing move''. The terminology originated with French-speaking Hex players, such as Mickaël Launay in [https://youtu.be/uAiU7hSiU5A this video], where it is called ''le coup du pêcheur'', literally "the fisherman's move". A fishing move is a kind of trap that is sometimes effective against beginners, but should not be used against experienced players, as it is usually bad for the player who makes it. A fishing move can also sometimes be a last ditch effort by a player who is losing and is desperate for the opponent to make a mistake.

Fishing moves often take the form of playing a [[peep]] in an ascending bridge. In the following example, a red 2nd row ladder is approaching from the left, and the blue bridge is ''ascending'' (relative to the direction of the ladder).
<hexboard size="5x7"
coords="none"
edges="bottom"
contents="R b2 a4 b4 B a5 b5 d3 f2"
/>
Red plays the fishing move 1, hoping that Blue will defend the bridge and Red will get a 2nd row ladder escape. Instead, Blue should foil at 2.
<hexboard size="5x7"
coords="none"
edges="bottom"
contents="R b3 c3 e1 B a5 b5 d3 f2 R 1:e3 B 2:d4 R 3:e2 B 4:e4"
/>
Note that if Red follows through on the bridge threat, the result is a 3rd row ladder for Red, which is typically worse than the 2nd row ladder Red would have gotten otherwise. Playing a fishing move in an ascending bridge usually results in raising the ladder by one row, and is bad for the player who plays it.

On the other hand, playing a peep in a ''descending'' bridge is often useful and not a fishing move. It typically serves to lower the ladder by one row (for example converting a 4th row ladder to a 3rd row ladder), or to escape a 2nd row ladder outright. Consider the following example, with a red 3rd row ladder approaching from the left. Note that the blue bridges are descending (relative to the direction of the ladder).
<hexboard size="5x9"
coords="none"
edges="bottom"
contents="R a3 b3 h1 i2 B a4 b4 d2 e2 f3 g3 h4"
/>
Red's intrusion 3 lowers the ladder from 3rd row to 2nd row, and Red's 9 escapes it. (Moves 4, 5, 10, and 11 are not usually played, but have been included for clarity).
<hexboard size="5x9"
coords="none"
edges="bottom"
contents="R a3 b3 h1 i2 B a4 b4 d2 e2 f3 g3 h4 R 1:c3 B 2:c4 R 3:e3 B 4:f2 R 5:d3 B 6:d5 R 7:e4 B 8:e5 R 9:g4 B 10:h3 R 11:f4"
/>

== References ==

* Mickaël Launay, ''Le Jeu de Hex, Tactique et Strategie - Niveau 1, [https://youtu.be/uAiU7hSiU5A 6. Le coup du pêcheur]'', 2014.

[[category:ladder]]
[[category:definition]]

User:Hexanna

2026-03-01T00:50:37Z

Hexanna: current projects

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata] is a GUI and CLI for analyzing Hex that I vibe-coded. It's still under active development, but I already use it as my main GUI.
* One of the GUI's main advantages (and a primary reason I created it) is that it's fast and responsive. It's not as useful for "traditional" analysis as some other GUI's — it doesn't support adding comments to moves, or branching move history.
* The CLI is primarily intended for AI coding tools and scripts. For example, I have used it to proofread and validate some of the KataHex eval claims made in my articles.

==current projects==

* working on Hexata GUI and CLI
* using Hexata CLI in vibe-coded scripts to understand Hex better
** this is harder than I anticipated; there are a lot of decisions to make and a lot of things that could go wrong
** for example: I've been trying to understand the strength of stones relative to local patterns in a more systematic way (removing bias introduced by the board edges and other sources of noise). I have always preferred using larger boards like 25×25 because the center of the board is far away from the edges, and katahex evals are further away from unstable 0% or 100% regions. I checked whether my methodology gave a consistent result across board sizes. unfortunately, it didn't, even though "the strength of stones relative to local patterns" is a concept that's theoretically independent of board size. preliminary investigation of this inconsistency shows that — despite the application of appropriate noise-reduction methods — there appears to be a systematic bias to its evals (on katahex 20240812) that becomes visible around the size 22→23 boundary and gets progressively worse as board size increases. so, many of my 25×25 analyses on local patterns are slightly wrong (luckily, I didn't use a larger size like 31 where the bias is more severe); a smaller size like 20–22 may be optimal for this net. (usual caveat: this finding is specific to katahex 20240812 and does not generalize to other nets. it does not imply that size 25 is categorically worse than sizes 20–22 for other kinds of analyses.)

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from [[KataHex]]==

* Swap map for 19×19 generated with KataHex net 20240812 using [https://github.com/hexanna1/hexata Hexata], with around 30k–40k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 84:(d3 p17)
19:(e3 o17)
108:(f3 n17)
118:(g3 m17)
107:(h3 l17)
127:(i3 k17)
140:(j3 j17)
136:(k3 i17)
140:(l3 h17)
72:(m3 g17)
13:(n3 f17)
73:(o3 e17)
70:(p3 d17)
142:(h4 l16)
86:(i4 k16)
76:(j4 j16)
92:(k4 i16)
104:(l4 h16)
165:(m4 g16)
158:(i5 k15)
124:(j5 j15)
161:(k5 i15)
127:(q2 c18)
214:(p2 d18)
136:(b17 r3)
250:(b18 r2)
121:(a2 s18)
118:(b2 r18)
89:(c2 q18)
220:(d2 p18)
127:(a3 s17)
184:(b3 r17)
199:(c3 q17)
126:(a4 s16)
115:(b4 r16)
204:(a5 s15)
146:(a6 s14)
136:(a7 s13)
195:(a8 s12)
167:(a9 s11)
87:(a10 s10)
126:(a11 s9)
168:(a12 s8)
111:(a13 s7)
92:(a14 s6)
53:(a15 s5)
73:(a16 s4)
164:(a17 s3)
242:(a18 s2)
70:(a19 s1)
523:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

Strategic advice from KataHex

2026-02-20T01:44:31Z

Hexanna: fix typo

==Introduction==

Several hex bots have popped up in the last few years. Many are neural nets and adaptations of Go bots like AlphaGo Zero. The most recent (and likely strongest) as of 2025 is [[KataHex]], which is based on the open-source KataGo. These bots have influenced human play in many aspects already; one of the most notable is the bots' preference for the 4-4 obtuse corner over the 5-5. Many old strategy guides have become partially (if not mostly) outdated.

This article attempts to bring some strategy advice "up-to-date". There's a lot of random insights; most are a direct result of playing around with KataHex evaluations for thousands of hours, including analyzing the games of strong players for common mistakes. Since this is strategic advice and not tactics, the advice won't apply 100% of the time, and many players may disagree with some parts. Win percentages and evaluations in this article are based on KataHex net [https://github.com/hzyhhzy/KataGo/releases/tag/Hex_20240812 20240812], which is significantly stronger than previous nets.

Care has been taken to only draw conclusions when there are multiple examples of KataHex preferring something. However, the main goal is to give actionable advice to make players stronger, and there isn't rigorous proof for basically any of these statements.

===Big picture, and some meta-points===

* '''TL;DR:''' What are the most important principles? Even if you learn nothing else, you should learn these:
*# Avoid playing close to your own edge, especially in the opening, unless you have a very good reason to do so.
*#* Think of your own edge as a whole row of stones of your color. You already have a bunch of stones (and thus strength) there, and playing yet another one close by would be a waste. That stone is much better served elsewhere, where you don't have as much strength.
*# Bridges are the most efficient way to connect your own stones—not adjacent stones, trapezoids, crescents, or hammocks. If there are no opposing stones nearby, playing adjacent to an existing stone of your color is usually a mistake; if you want to do that, consider whether there's a better move a bridge away from that stone.
*#* If there are nearby opposing stones, things become more nuanced, but bridging to your existing stones is often still a good idea.
*# Usually, cut through an opponent's bridge if given the opportunity, and avoid letting it happen to you. In the pattern below, Red cuts through Blue's two stones a bridge apart. This is very good for Red and bad for Blue, because it's very hard for Blue to connect the two stones now.
*#: <hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 c1 R b1 1:b2" />
* This guide assumes you've read [http://www.mseymour.ca/hex_book/hexstrat.html ''Hex: A Strategy Guide''] or have equivalent knowledge.
** ''Hex: A Strategy Guide'' predates KataHex, but it's very well made, and 90% of the advice is still good.
* On smaller boards, tactics and raw calculation matter more. On larger boards, strategy matters more. (It's ultimately a spectrum, and it varies by skill level: for beginners, size 13 might be considered large; for strong players, tactics can matter a great deal on size 13.)
** On smaller boards, the exact board size affects strategy (some moves can be good on 11×11 but not 12×12). The larger the board, the less the exact board size matters. The general strategy for 19×19 and 20×20 should be nearly identical.
** On smaller boards, there is more space close to corners and edges. On larger boards, there is more "open" space (hexes not close to corners and edges). I intentionally said "open" space and not "center" space, because there is not much special about hexes near the center of the board, except that they are far away from corners and edges. But on large boards, there are many such areas.
** In particular, the larger the board size (starting at around size 13–15), the more important it is to know how to play in "open" areas. Many existing guides cover corner/edge strategy heavily—and correctly so, as corners/edges remain important up to very large sizes (25–30 or so). People read those guides and often know how to play the first 10–20 moves of a 17×17 game, but often not after that, when most of the game is still remaining!
** Strategy can also be ''local'' or ''global''. Local strategy tells you how to play within a small region of the board, and global strategy tells you how to play in relation to the whole board (or across multiple local regions). Both are important.
** The strategy of "open" areas is mostly local, and so the larger the board, the higher the relative importance of local strategy. Why?
*** On a large Hex board, the "middle" is relatively homogeneous—two separate open areas near the middle are basically the same in terms of how you should play them; it doesn't matter that one of the areas is 15 hexes away from an edge and the other one 18. Because of this homogeneity, the same basic principles will apply multiple times in the span of a game.
*** Knowing how to play across multiple local regions is very hard, and it's often different across games (especially for humans). It's better to master local play first, since similar local play principles will be relevant every game.
* How should a game of Hex (on larger boards) broadly feel like?
** While the win condition is binary (you either connect or you don't), the actual gameplay feels like a gradual accumulation of tiny advantages and disadvantages. Strong players often "know" they're winning in the middlegame, not because they see an actual way to connect, but because their opponent made some mistakes, however small, along the way. (You rarely know for sure who's winning until the end, and you're ultimately making an educated guess.)
** In particular, it's usually very hard to tell who's winning just by looking at a position. Contrast that with games like chess, where you can often tell by counting up material and broadly looking at positional factors. In Hex, it's much more tractable to tell how good your position is ''relative to a few moves ago.''
** Relatedly, there's (almost) no meaningful notion of "long-term strategy" in Hex. If anything, it can be harmful to create a long-term plan and stubbornly stick to it, because the moves you should play often differ greatly depending on your opponent's responses.
** Hex strategy is equally what to play and what ''not'' to play. The latter is frequently underemphasized. Each stone is worth a lot, and even one bad move can "undo" 10–20 good ones. The risk-reward balance strongly favors avoiding mistakes, rather than attempting risky but potentially "brilliant" moves.
* Now some points about how to learn. From what I observe, people often read strategy advice and then fail to follow it consistently. This is often either because of skepticism, or not realizing just how often particular pieces of advice should be followed (it's tempting to think that you've found the exception to the rule).
** It's healthy to be skeptical—there is a lot of ''bad'' strategy advice, and it would probably do more harm than good to implement every piece of advice you come across. So I'm not expecting readers to take everything in this guide at face value either (and many statements have nuances/exceptions, some of which I don't even realize should be explained to readers; it's hard to translate precise intuitions into words after all). Unfortunately, separating the signal from the noise is hard. If you're skeptical about a principle or how strongly it holds, try running it through KataHex yourself (if you have it installed), and try variations to understand the nuances, when the principle holds, and importantly ''why'' it holds. Sometimes people are skeptical of KataHex itself or think they can do better; I think there are good heuristic reasons why this thinking is misguided (the most simple one is that KataHex is far stronger than the strongest humans).

==General==

===Common mistake: playing near your own edge===

Playing near your own edge with no nearby stones is a bad idea in the opening and even middlegame. This is one of the most common mistakes among human players. You need a strong (often tactical) reason to do otherwise.
* When is playing near your own edge a good idea? There are a few cases, like when you're playing adjacent to existing stones, responding to a joseki, or pushing a ladder as the attacker, but all of these involve stones in the vicinity.
* This holds most strongly for 13×13 to 19×19, but applies more often than not to 11×11 too.
* Playing near your own edge in the opening or early middlegame should be a "last resort". I would recommend never doing so in the first 30-40 moves in 19×19 unless you have a very good reason.

===Common mistake: bad minimaxing===

Many intermediate and strong players are too eager to minimax. They play bad minimaxing moves (that KataHex thinks are inferior to directly connecting) more often than they fail to minimax. The issue is that minimaxing isn't always free; your opponent can often intrude for useful territory, and sometimes the territory your opponent gains is more useful than what you gain by minimaxing. It's a difficult skill to judge which player gains more from a minimax, but there are common patterns.
* Players frequently attempt to minimax by playing adjacent to their existing stone on the 5th or 6th row of their own edge. Playing two adjacent stones like this is not very efficient and typically a mistake.
* Blue's minimaxing attempt is unsuccessful; KataHex already thinks Red is 98% to win: https://hexworld.org/board/#15nc1,c2e11c13e10b12e9c10d8b9c7a8b6c6b7d7c8e8d9
* A common example; Red is 98% to win: https://hexworld.org/board/#13nc1,c2d10c11c10b11c9b8a10. Note that Red 7 at b8 is stronger than at a10, because b8 combines well with c2 and b5 on 13×13. On 15×15, KataHex prefers 7. a12 (the equivalent of a10 on 13×13) instead.
* A case where minimaxing is in fact the best move: https://hexworld.org/board/#13nc1,c2d10c9d11

===Common mistake: bad bridge peep===

A common scenario is the [[Peep#Bridge_peep | bridge peep]]. Blue should play at (*) as long as she thinks it's forcing enough for Red to defend his bridge.

However, even if Blue hasn't played (*) yet, Red should usually mentally place a blue stone there. Especially on larger boards, Blue will be able to play at (*) sometime in the future. This is important: even if (*) is empty, I like to think of this pattern as ''implying'' a blue stone at (*) (in the absence of tactics that somehow prevent Blue from playing there). Another way (useful for me, but the analogy might be nonsense to others) to think about it is that Blue's stone at A is well-placed enough relative to Red's bridge that she has some "potential energy" at (*).

<hexboard size="3x3" coords="hide" edges="hide" contents="R c1 b3 B A:a2 E *:c2" />

However, 80% of the time, if there's no blue stone at A, Blue should not intrude on either side of the bridge. Intruding is usually a mistake because it settles the [[question]] for Red. If Red knows he will have a stone at A or B because Blue already intruded in the other spot, Red can better plan his future moves because of that! There are exceptions; if Blue is confident that only one of the intrusions will ever be useful for her, then intruding is often strong.

<hexboard size="3x2" coords="hide" edges="hide" contents="R b1 a3 E A:a2 B:b2"/>

===Ladders===
* Don't worry about switchbacks, climbing, or ladder creation templates on 15x15 or larger. They can be theoretically interesting and make for good puzzles, but games are rarely close enough for these to matter. I personally don't think about them on any board size, but others may reasonably disagree with this approach.
* Q: When should the defender of a ladder yield? A: Usually, don't yield unless your opponent can cut through two of your stones, one above and one below the ladder. In that case, consider yielding.
** [https://hexworld.org/board/#13nc1,c2e9c10d8b9c11d10d12 Here] is an example. If Blue instead plays 8. d11, Red 9. e10 cuts through e9 and d11.
** In [https://playhex.org/games/66b8a125-bf02-42d2-8900-af3042a8ccf8 this game] from the [[Tournaments#Hex_Monthly | Hex Monthly 21]] ([https://hexworld.org/board/#14c1,c2d11k10k4b5f10g9i9i8g10h9h10j8j9b12c10l8k8l7k7l6k11j11h12m5k6l4m4l5l9k9h13i13l3k5i5j5j4i6h6f8h7i7f9g8f7e8e7d8d7b8c8b9c9a11b13d10c11c12,c13d12d13e11:rw HexWorld link]), the yielding move 60. c14 is winning, while the played 60. c13 (which allows Red d12, cutting through Blue's c13 and d11 stones) is losing.

===Other===

* Be aggressive; always be ready to capitalize on an opponent's mistake. Here are a few illustrative examples of this and some of the above concepts:
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2368410&nmove=8 this game], 6. c9 was a mistake (and an example of incorrect minimaxing); 7. a10 is the [[Openings_on_19_x_19#4-4_obtuse_corner | standard reply]]. White should have played 8. b8; j4 was a mistake. Black's 9. b8 is the "capitalizing" move; KataHex already says 99% win rate for Black. In this case, there were several good options on move 9. These alternatives are threatening enough that Black didn't have to play b8, but this won't always be true, like in the next example.
** [https://littlegolem.net/jsp/game/game.jsp?gid=2362913&nmove=26 Here] is another example (interestingly, White resigned in a winning position according to the bot, though the final position is highly tactical). KataHex actually thinks Black has only an 35% win rate after 26. m6. However, 27. d11 is a "minimaxing with adjacent stones" mistake, bringing Black's win rate down to 3%.
** [https://www.littlegolem.net/jsp/game/game.jsp?gid=2400947&nmove=6 Here] is an example where White's win rate increased consistently from 65% after move 6 to 98% after move 16. The aggressive bridging sequence in moves 8–16 is very strong because it basically neutralizes Black's stones on d16 and d17; it is a big mistake for Black to ignore the threat here. Move 11 is also an example of a bad bridge peep.

* Don't play a move that makes your opponent's existing stones unnecessarily well-placed relative to your new stone. This is covered briefly in a couple places in the [[Openings_on_19_x_19#General_principles | 19×19 opening guide]]. Here are a couple more examples.
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2369954&nmove=26 this game] ([https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6 HexWorld link]), what should Black play on move 27? White is threatening to connect 26 back to the 16–24 group if Black isn't careful. Black's obvious options are to play at c8 and c9. Which option is better?
** A tempting answer is c8 because it appears to connect more closely to the top. However, note that White's stone on move 14 functions exactly like an opening a3 stone. For example, it can escape 2nd but not 3rd row ladders. If Black plays c8, White can play the [[A3 escape trick]], as shown in [https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6c8e7d8e8d9e9 this continuation]. This sequence allows White to "make the most" of stone 14, and Black shouldn't allow it. Better for Black is 27. c9, which doesn't allow White to carry out a 3rd-to-5th row switchback. Indeed, KataHex evaluates c9 as 97% to win for Black and c8 as only 30%.
** Consider [https://hexworld.org/board/#13n,c2d10c11c10b11e6b10d8l4l3c8d5b5d6h7h6j5i6k7 this position] (which is based on [https://www.littlegolem.net/jsp/game/game.jsp?gid=2369782 this game]). Where should White play in the lower-right acute corner? Black had just played at the "3-7" point from White's perspective; as mentioned in the 19×19 opening guide, a strong response to 3-7 is 4-4 (at j10) instead of the usual 5-4 (at i10). A player who is short on time can play j10 without thinking too hard. As it turns out, KataHex thinks j10 is the best move, evaluating it as 91% for White, whereas j9 is only around 50%.

* Sometimes, you know you have a completely winning position after your opponent blunders very early in the game. Your goal is to preserve that win until the end of the game. Don't overextend yourself and try to win too "quickly", possibly making suboptimal moves in the process. Even when KataHex plays itself with a significant handicap (like playing without swap), a significant fraction of the board seems necessary to carry out the win, assuming strong defense. I prefer taking my time to "fill the board" by focusing on playing moves at least as good as my opponent's, instead of worrying about connecting in as few moves as possible.

==Local patterns==

An especially effective way to improve at Hex is to learn and gain intuition about lots of small local patterns. These patterns are small enough that they occur basically everywhere when you play Hex. Knowing the patterns can help you avoid blunders (for example, avoiding adjacent or bridge ''allow-cuts'', as explained below) and make good moves when it's otherwise unclear what to play.

===Cuts===
<hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 c1 R b1 1:b2" />
<hexboard size="1x3" float="inline" edges="none" coords="none" contents="B a1 c1 R 1:b1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" visible="-(a1 b1 d1 a3 c3 d3)" contents="B a2 d2 R c1 1:b3 S b2 c2 E *:(b2 c2)" />
<hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a3 c1)" contents="B a1 c3 R 1:b2 S b1 a2 c2 b3" />
Consider the four patterns above. Let's call the patterns '''adjacent cut''', '''near cut''', '''short bridge cut''' (or just '''bridge cut'''), and '''long bridge cut''' respectively. In some sense, Red is "cutting" through two of Blue's stones in each pattern, so Red is the '''attacker''' and Blue is the '''defender'''. We can also use '''cut''' as an attacking verb and '''allow-cut''' as a defending verb. For example, the adjacent cut pattern is favorable for the attacker (Red above), so we can say that it's a good idea to ''adjacent-cut'' through opponent stones, and a bad idea to ''adjacent allow-cut'' (meaning it's a bad idea to play a move that allows your opponent to adjacent-cut through two of your stones).

Why is it strong to adjacent-cut (and equivalently, why is it a mistake to adjacent allow-cut)? The most intuitive reason is that if Blue allow-cuts, it takes 3 extra moves for her to connect her two stones, as shown below. In a sense, this makes it unlikely for both stones to be concurrently useful in the final connection — if one of the stones proves useful, chances are the other one won't be.
<hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 c3)" contents="B a2 c1 1:a3 2:b3 3:c2 R b1 b2" />

The (short) bridge cut is strong for the attacker (Red above) because he captures the two hexes marked (*). If Blue allow-cuts, often the damage has already been done even if Red doesn't actually cut through immediately on the next turn. (However, note that this isn't always true; some tactical situations require Red to cut through immediately instead of playing elsewhere.)
* There are situations where Blue ''should'' allow-cut. If Red's cutting move comes with concessions, like the move is on his second or third row (and especially if Red already has a stone near his own edge, so cutting through would be overplaying that edge), the pattern is much less favorable to Red.

The near cut and long bridge cut are also favorable for the attacker, but much less clearly so — there are many more "exceptions" where cutting is bad and allow-cutting is fine. On the other hand, the adjacent and short bridge cuts are quite strong, and cutting through is usually the best move (unless doing so involves playing a stone that's too close to your own edge, or another major concession).

Note that allow-cuts can occur not only ''proactively'' before the attacker plays the cutting move, but also ''retroactively'' after the "cutting" move is played (of course, it's not really a "cutting" move until after the defender allow-cuts). In other words, Blue 1 is considered an allow-cut in both of these examples (proactive on the left and retroactive on the right), and they are equally bad for Blue assuming that Red intends to cut in the proactive case:
<hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 1:c1 R b1" />
<hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 1:c1 R b1 b2" />

An important way of pruning moves and avoiding blunders is making sure you don't play moves that adjacent or bridge allow-cut, either proactively or retroactively, unless you have a good reason to do so. In particular, the retroactive case can be easy to overlook if you're not used to thinking in this way.

===Small patterns===
<hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="B b2 R b1 1:a2" />
<hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a3 c2 c3)" contents="B b3 R c1 1:a2" />
<hexboard size="4x2" float="inline" edges="none" coords="none" visible="-(a1 b4)" contents="B a4 R 1:b1" />

In each of these patterns, we will discuss the strength of Red's move 1 ''in a vacuum'', meaning in the absence of nearby stones, like near the middle of an empty board on 19×19 or even larger boards. It is most useful to analyze each pattern without the influence of nearby stones or edges (which can be thought of as virtual rows of stones).
* Pattern 1: It's typically an inaccuracy for Blue to play adjacent to a red stone in a vacuum. Red's best response is usually playing adjacent to both stones, as in Red 1.
* Pattern 2: Blue plays a bridge away from Red's stone. This is often a good move, and Red 1 a bridge away from both stones is also good.
* Pattern 3: Red 1 is a classic block. This is often fine for Red, and Blue doesn't have an obvious best local response. Blue could tenuki here.

===Capture patterns===

Know the most common [[captured cell]] patterns. It really reduces the number of moves you have to consider.

The bridge cut is one of the most important capture patterns.

* A typical example can be found in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2233736&nmove=10 this game]. After 10. g11, Black would have the easiest time with 11. h9, with 95% win rate, instead of the played k8 with 74% win rate. After k8 was played, one of White's best moves would've been h9 to prevent Black from playing the same. KataHex does think Black has other reasonable options on move 11, but they mostly appear to just delay h9.

Here is another common capture pattern. The other patterns on [[captured cell]] are useless because they have too many Red stones adjacent to each other, which is too inefficient to occur in normal play.

<hexboard size="3x3" coords="hide" edges="hide" contents="R c1 c3 b3 B a2 S b2 c2 E *:(b2 c2)" />

This ''one-sided'' capture pattern is very common. I like to mentally replace A with a blue stone and (*) with a red stone (even though Blue won't always achieve this), because Blue should never play at (*). [[Edge template IV3a]] is useful to know here.
<hexboard size="3x2" coords="hide" edges="hide" visible="-a1" contents="R b1 a3 b3 S a2 b2 E *:b2 A:a2" />

The trapezoid and crescent also one-sided-capture the hexes at (*). Anecdotally, this fact is more useful on smaller boards like 11×11, but it's less common overall than the previous capture patterns.
<hexboard size="3x3" float="inline" coords="none" edges="none" visible="-a1" contents="R b1 a3 c3 c1 S a2--c2 b3 E *:(c2 b3)" />
<hexboard size="3x3" float="inline" coords="none" edges="none" visible="-a1" contents="R a2 b1 c1 c3 S b2 c2 a3 b3 E *:c2" />

==How to think about the opening and middlegame==

Some of these points sound like hyperbole, but they are a fairly accurate representation of how I think about Hex, especially on larger boards. There are exceptions, but exceptions are rarer than one might think.

* In the first half of the game, focus all of your attention into playing "efficient" moves. These are moves that work well with existing stones and board edges. Excellent intuition of efficient patterns is enough to get strong middlegame positions against even top players (as of 2024).
* Don't worry about tactics until the late middlegame. It's a waste of time to analyze sequences >10 moves deep. In the opening, the best move you can find with only intuition is no worse than the best move you can find with calculation.
** Why? Compared to bots, humans are very bad at calculation. Your intuition (which you learn directly from KataHex) is more useful than hypothetical hours or days of error-prone human analysis.
* Use your knowledge of efficient patterns to plan ahead. This is where you want to do shallow analysis, typically 1-5 moves deep. A good move does as much as possible:
** allows you to play efficient moves later
** prevents your opponent from playing an efficient move outright
** allows your opponent an efficient move, but only with concessions (say, a locally efficient but globally inefficient move, or a move that's efficient with relative to one stone but inefficient relative to another)

These points require a lot of elaboration. What do I mean by "efficient"? (Not to be confused with [[efficiency]].) There are two broad notions, '''global''' and '''local''' efficiency.
* Globally efficient stone: Broadly, any move that isn't too close to your own edge. Moves that are strong in the absence of nearby stones (like on an empty board). Corner moves are very efficient. On 19×19, moves near your opponent's 5th row are very efficient.
* Locally efficient stone: In the presence of nearby stones, some configurations are broadly "good" or "bad" for one player. They aren't ''always'' good or bad because of tactics, but because humans can't calculate everything, it's extremely useful to have decent heuristics that work most of the time.

In a sense, you want to play the move that maximizes the "sum" of local and global efficiency. This isn't an exact science; use your intuition to make an educated guess.
* The best move locally is rarely the best move globally. Hex is a game of concessions and tradeoffs. There is no "free lunch," unless you count concepts like inferiority/domination.
* Conversely, if it looks like your opponent is beating you on one side of the board, there are two possibilities:
*# You played suboptimal moves in that region.
*# Your opponent overplayed that region, which necessarily means they made concessions elsewhere (usually the opposite side of the board). This means your opponent ''underplayed'' the other region.
* If option 1 is true, you are probably losing anyways, so assume (or hope) option 2 is true. If you opponent overplays region A and underplays region B, it's typically a good idea to focus your play in region B. Why? If you keep playing in region A, you're potentially helping your opponent out of their mistake, by making your opponent less over-connected in region A. For example, your opponent could have a minimaxing move that makes them much stronger in region B, at the "concession" of making them weaker in region A — but this isn't really a concession because your opponent had strength to spare in region A.

The discussion so far has been highly abstract. What does efficient play look like in practice? As explained earlier, it's strong to adjacent-cut and bridge-cut through your opponent's stones. The small patterns in the "Capture patterns" section above are also locally efficient for the capturing player. Of course, just as offense and defense are equivalent in Hex, so are playing locally efficient moves and inducing your opponent to play locally ''inefficient'' moves.

In the following examples, P1 and P2 are efficient patterns for Red relative the corresponding edges, while P3 is inefficient. P1 is common when Red opens with a first column opening. P2 requires elaboration: normally Red 1 is a weak move because it's too close to Red's own edge. However, when Red must play on his first 3 rows (because of very limited space), Red 1 or its mirror image at (*) is usually the best, because it prevents Blue from playing there and forming the equivalent of P1 herself. P2 also occurs in theoretical contexts, like [[Template Va#If Blue moves at y:]]. In P3, Red 1 is a tempting (because of [[edge template III1b]]), common, but inefficient move, because Blue's response at one of (*) is usually strong. If Red must play near his own edge, he should instead play as in P2 above.
P1: <hexboard size="3x2" float="inline" edges="left" coords="none" contents="R a2 1:b3" />
P2: <hexboard size="2x6" float="inline" edges="bottom" coords="none" contents="B c2 R 1:e1 E *:b1" />
P3: <hexboard size="3x5" float="inline" edges="bottom" coords="none" contents="B c3 R 1:d1 E *:(b2 e2)" />

===Making inferences: An extended example===

Use the patterns and intuitions that you know, to reason and make inferences in unfamiliar positions. You might not always find the best move, but you can often prune bad moves. Here is an extended example, involving several closely related patterns that are useful to know in isolation.
# ''Pattern:'' If you play the 4-4 3-3 obtuse corner joseki, it is important to know that Blue 6 [https://hexworld.org/board/#19nc1,d16c17d17c18d18e14c15 here] is weak.
#* This is because Red 7 undermines Blue 6 and also reduces Blue 2/4 from a 3rd row ladder escape to a 2nd row escape. Also, Blue's potential intrusion at c16 doesn't gain her anything.
#* Instead, [https://hexworld.org/board/#19nc1,d16c17d17c18d18d15 here] is a good local response for Blue. This gives Blue a 5th row ladder escape while blocking Red 1.
#* On the other hand, if Blue only has a 2nd row escape and doesn't have the "potential energy" to get a 3rd row escape, you might guess that Blue 6 above becomes a good local move, because Red 7 loses one of its main strengths. Such a guess would be correct: in [https://hexworld.org/board/#19nc1,d16c17p15d17e16g16e17f18d18e14c15 this position], Blue 10 is a strong move. On the other hand, Blue 10 at d15 would be weak here, as it's quite wasteful, only converting a 2nd row escape into a 3rd row escape.
# ''Pattern:'' If Red tenukis on move 3, then Blue 4 is locally strong in [https://hexworld.org/board/#19nc1,d16e14:pf15 this position]. In particular, Blue 4 is much stronger than bridging directly towards Blue's own edge with g13, because the former is a strong blocking move that reduces the effectiveness of Red 1. This strength is in spite of Red's intrusion on move 5. (I first saw this move in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2090091&nmove=16 this bot game] from [https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=739 this thread on LittleGolem].)
#* Variations of this pattern appear frequently in obtuse corner joseki. If Red started with the 5-5 obtuse corner, then the equivalent Blue 4 is [https://hexworld.org/board/#19nc1,e15f13:pg14 also] strong.
#* Another simple example is Red 5 in [https://hexworld.org/board/#19nc1,d16d15f14:pe13 this position].
# ''Inference:'' Let us consider [https://hexworld.org/board/#19nc1,d16c17d17c18d18:p this position] again. Suppose Blue tenukis on move 6. What is a good local response for Red on move 7? There is more than one right answer, but you just need to find one move that isn't a blunder.
#* ''Hint:'' Use the two patterns above to come up with an answer.
#* ''Hint:'' Consider that Blue has a 3rd row ladder escape, so Blue d15 is locally strong but Blue e14 is weak. This is the case for future blue stones in this local region, even though Blue already played elsewhere on move 6. Therefore, Red should perhaps try to play a move that makes Blue d15 less attractive, even if it comes at the concession of making Blue e14 more attractive (since the latter is ''a priori'' a weak move, such a concession gives up less than it otherwise might).
#* ''Answer:'' Red 7 [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13 here] is a good local move.
#** Why? Ignoring stones 2-5, the presence of Red 1 and 7 makes Blue e14 more attractive. Red's 4-4 stone, much like the obtuse corner opening, tends to shift Blue's efficient stone locations up one row. With that in mind, Red's move 7 is a bit like Red e14 on an empty board, where Blue e15 is [https://hexworld.org/board/#19nc1,e14e15 the best response]. However, stones 2-5 make Blue e14 a bad move, despite Red 7 making it slightly less bad (this is the concession-but-not-really-a-concession that Red intended).
#** Let's now consider Blue d15. Normally a decent local move after stones 1-5, it becomes quite weak after Red 7, because of [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13d15f14 this sequence]. This is a typical example of the second pattern in this extended example. (Actually, if Blue had two moves in a row in this region, Blue 8 wouldn't be so bad because she could follow up with f14, where Red wants to play move 9. This is a tactical situation where it's bad for Red to tenuki.)
#** If we wanted to be complete, Blue actually has another option for move 8, f14 first which threatens d15: [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13f14d15d14e14 here]. This is an example of the "skew cut" from above. Red should respond with d15 himself, or else Blue can play d15. This is probably Blue's best local response.
#** Every move makes a concession, but some concessions are more effective than others. As alluded to in the hint, Red 7 weakens a strong Blue reply (d15) while strengthening a weak Blue reply (e14). Red's goal is to minimize the strength of Blue's ''best'' response (rather than the strength of Blue's average response), and Red 7 at e13 accomplishes that beautifully.

Part of getting stronger at Hex (as with other games) is being able to compress knowledge and patterns effectively. If you understand this extended example, you can infer the best move in many similar but novel situations, without memorizing each one individually. If you forget which move is exactly the best, you can recover the right move with high probability by just reasoning about related patterns that you do know are strong or weak. The "A3 escape trick" example earlier in this article is another instance of this concept.

On larger boards and particularly near the opening or middlegame, when you have a lot of time to think about a position, it may be worth spending a good chunk of it making inferences and scoping out moves that are either good or to be avoided by each side, rather than spending most of your time calculating tons of lines and variations.
* In fact, once you've scoped out that "moves A and B are good for Red, while moves X and Y are bad for Red" and similarly for Blue, that makes the process of analyzing and pruning lines much more efficient.

==Joseki==

* The 4-3 corner move is almost always a mistake compared to the 5-4, even on 11×11. If your opponent plays 4-3, 90% of the time the best local reply is 3-3. If 4-3 is played, here are some common sequences:
: <hexboard size="5x7" float="inline" coords="hide" edges="bottom right" contents="R 1:e2 B 2:e3 R 3:d3"
/> <hexboard size="5x7" float="inline" coords="hide" edges="bottom right" contents="R 1:e2 B 2:e3 R 3:f2 B 4:f3 R 5:d3 E a:b4 b:c5"
/>
: In the second joseki, Blue should play either (a) or (b), or she can defer the question until later. On larger boards with no nearby stones, (b) is often better.
* If your opponent plays the 5-4 joseki, a reasonable choice is to always play the 4-4 "high intrusion." KataHex prefers the 4-4 response by far, especially on 13×13 and larger. It's easily the "safest" choice, and many strong players have recently shown a clear preference for 4-4.
* In this often-played joseki, Red 7 is a mistake because of the territory Blue gets after move 8:
: <hexboard size="6x10" float="inline" edges="bottom right" coords="none" contents="R 1:g2 B 2:g3 R 3:h2 B 4:h4 R 5:i3 B 6:i4 R 7:f3 B 8:e5 R 9:d4 B 10:c6 R 11:b5 B 12:c4 R 13:c5" />
:* If the obtuse corner on Red's same edge is empty, a more principled response for Red is to play in that corner, like this:
:: <hexboard size="6x14" float="inline" edges="bottom left right" coords="none" contents="R 1:k2 B 2:k3 R 3:l2 B 4:l4 R 5:d3" />
:* If the obtuse corner is already occupied by Blue, then Red should still respond in the obtuse corner. One reasonable option (among others) is this:
:: <hexboard size="6x14" float="inline" edges="bottom left right" coords="none" contents="B d3 R 1:k2 B 2:k3 R 3:l2 B 4:l4 R 5:e3" />
* See more discussion of acute corner joseki at [[Openings_on_19_x_19#Acute_corner_theory]]. For the obtuse corner, see [[Openings_on_19_x_19#Obtuse_corner_theory]] which covers responses to the 4-4 and 5-5 obtuse corner in great detail. Most of this is applicable to 13×13 and larger, and not just 19×19.

==Specific tactics==
===Threatening a bridge cut===
One of my favorite tactics is threatening a bridge cut, where the opponent needs to play too close to their own edge to defend:
<hexboard size="8x8" float="inline" edges="bottom left" coords="hide" contents="B *:d5 g2 R 1:f4 B 2:e3" />
Red 1 is a strong move, because it forces Blue to respond at 2 (or else Red will play the bridge cut there), but Blue 2 is normally overplaying on the left side when combined with Blue's stone at (*). Be on the lookout for this when your opponent plays the 4-4 and 7-7 obtuse corner, or 5-5 and 8-8.

This tactic can work well in other configurations, like [https://hexworld.org/board/#19c1,:pg8:pg11h9 this] and [https://hexworld.org/board/#19c1,:pe9:ph9g8 this]. However, be careful when using this tactic! When Blue isn't actually making a concession by bridging her two stones, like when the pattern occurs further from her own edge, it can be a blunder for Red to threaten a bridge cut. [https://hexworld.org/board/#19nc1,:pg8:pj5i7h6 Here] is an example where Red 5 is a blunder in the absence of nearby stones.

[https://www.littlegolem.net/jsp/game/game.jsp?gid=2436000&nmove=12 Here] is an example from a real game. Black's best move is 13. h5 (in the actual game, Black doesn't play this until move 19 but is still arguably rewarded for doing so). One might think that h5 is too strong near the top edge because of c2 and b5, but this is not the case because the white stone on c5 partially neutralizes b5.

[[category:strategy]]

User:Hexanna

2026-01-31T16:05:07Z

Hexanna: update elo-based swap map

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata GUI]

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from [[KataHex]]==

* Swap map for 19×19 generated with KataHex net 20240812 using [https://github.com/hexanna1/hexata Hexata], with around 30k–40k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 84:(d3 p17)
19:(e3 o17)
108:(f3 n17)
118:(g3 m17)
107:(h3 l17)
127:(i3 k17)
140:(j3 j17)
136:(k3 i17)
140:(l3 h17)
72:(m3 g17)
13:(n3 f17)
73:(o3 e17)
70:(p3 d17)
142:(h4 l16)
86:(i4 k16)
76:(j4 j16)
92:(k4 i16)
104:(l4 h16)
165:(m4 g16)
158:(i5 k15)
124:(j5 j15)
161:(k5 i15)
127:(q2 c18)
214:(p2 d18)
136:(b17 r3)
250:(b18 r2)
121:(a2 s18)
118:(b2 r18)
89:(c2 q18)
220:(d2 p18)
127:(a3 s17)
184:(b3 r17)
199:(c3 q17)
126:(a4 s16)
115:(b4 r16)
204:(a5 s15)
146:(a6 s14)
136:(a7 s13)
195:(a8 s12)
167:(a9 s11)
87:(a10 s10)
126:(a11 s9)
168:(a12 s8)
111:(a13 s7)
92:(a14 s6)
53:(a15 s5)
73:(a16 s4)
164:(a17 s3)
242:(a18 s2)
70:(a19 s1)
523:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2026-01-31T15:43:46Z

Hexanna: new lightweight GUI

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[https://github.com/hexanna1/hexata Hexata GUI]

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-20T03:56:57Z

Hexanna:

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

* [part 2: talk about more complex patterns and other heuristics?]

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

* [part 3: talk about the edges, simple edge patterns, and how opposing stones influence the strength of your stones, both proactively and retroactively?]

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-20T03:47:00Z

Hexanna: more

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(g) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.

That was a pretty dense section, so let's take a breather and summarize.
* The main lesson is learning ''how to think'' when playing Hex — how to convert your knowledge into a useful assessment of a position. In real games, you won't be doing arithmetic, but you will have to weigh various tradeoffs intuitively.
** Here's an example of what that might actually look like during a game. In [https://hexworld.org/board/#14nc1,c2k4d11g6i5,h4j3 this position], Blue is considering whether to play 6. h4, which is a bridge allow-cut that allows Red to play 7. j3. One good way for Blue to think about it: "Should I bridge allow-cut here? Normally no, bridge allow-cuts are terrible, but in this case, their cutting move j3 would be on their third row, which is really bad for them. But I know that's not necessarily enough to make the allow-cut convincing, and it would still be too risky. But wait, in this case, they already played c2, making j3 even more redundant and overplaying the top edge. And h4 is locally strong too, at least considering the two-stone pattern of moves 4 and 5. Okay, all things considered, there's probably enough evidence to believe that h4 is a decent move here." Of course, you could apply similar reasoning when thinking about your opponent's potential replies (or further down the game tree).
* The main takeaways from the simple patterns:
** Think twice before playing adjacent to your own stone, unless both stones are adjacent to the same opponent stone. People frequently break this rule (by playing adjacent to their own stone without opposing stones) while trying to "minimax"; while this is occasionally strong (and you can learn the cases where it's good), it's usually a blunder. Generally, only analyze these moves after you've considered other options.
** On the other hand, playing a bridge away from your own stone is frequently strong — such moves should generally be on your radar.
** If you play adjacent or a bridge away from an opponent stone, you should expect an increased chance that the next few moves are all in the vicinity. That helps with pruning and deciding which lines to analyze more deeply.

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-19T03:25:06Z

Hexanna: clarify

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="B S:a3 E a:b3 b:c2 *:(c3 d2 d3 e1--e3)" />
::* Playing adjacent to S or a bridge away from S is a very slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. Moves further away from S, like those marked (*), are locally strong for Red. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(e) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-19T03:13:43Z

Hexanna: use diagrams in 1-stone case

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns and how to think about them===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move — if you only study patterns in the context of real games or on smaller boards, it's difficult to tell whether a move was good because the resulting pattern was inherently good, or because the rest of the board worked in your favor. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Suppose it's Red's turn in the below diagram.
:: <hexboard size="3x5" float="inline" edges="none" coords="none" visible="-(a1--d1 a2--b2)" contents="R S:a3 E a:b3 b:c2 c:c3 d:d2 e:d3 f:e1 g:e2 h:e3" />
::* Most choices, like playing a bridge away from S at (b), a "classic block" away at (d), or further away at (e)/(g)/(h), are locally strong moves. One major exception is playing adjacent to S, like at (a), which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps away at (c) or two bridge-moves away at (f) is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. We can use KataHex evals (applying some averaging and heuristic "noise reduction" methods) to quantify each move. For example, playing at (a) is a roughly 0.35-stone mistake in a vacuum, playing (c) or (f) is a 0.1-stone mistake because of the allow-cuts, and everything else — including tenuki — is pretty close to optimal.
::* ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
:* Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Suppose it's Red to move again.
:: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1--c1 a2--b2) c3" contents="B S:a3 E a:b3 b:c2" />
::* Playing adjacent to S or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. All other moves are almost equally good. KataHex says the adjacent reply at (a) is a 0.05-stone mistake and the bridge reply at (b) is a 0.1-stone mistake. Other moves, including tenuki, are close to optimal.
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(e) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

Strategic advice from KataHex

2025-12-18T04:31:32Z

Hexanna: typo

==Introduction==

Several hex bots have popped up in the last few years. Many are neural nets and adaptations of Go bots like AlphaGo Zero. The most recent (and likely strongest) as of 2025 is [[KataHex]], which is based on the open-source KataGo. These bots have influenced human play in many aspects already; one of the most notable is the bots' preference for the 4-4 obtuse corner over the 5-5. Many old strategy guides have become partially (if not mostly) outdated.

This article attempts to bring some strategy advice "up-to-date". There's a lot of random insights; most are a direct result of playing around with KataHex evaluations for thousands of hours, including analyzing the games of strong players for common mistakes. Since this is strategic advice and not tactics, the advice won't apply 100% of the time, and many players may disagree with some parts. Win percentages and evaluations in this article are based on KataHex net [https://github.com/hzyhhzy/KataGo/releases/tag/Hex_20240812 20240812], which is significantly stronger than previous nets.

Care has been taken to only draw conclusions when there are multiple examples of KataHex preferring something. However, the main goal is to give actionable advice to make players stronger, and there isn't rigorous proof for basically any of these statements.

===Big picture, and some meta-points===

* '''TL;DR:''' What are the most important principles? Even if you learn nothing else, you should learn these:
*# Avoid playing close to your own edge, especially in the opening, unless you have a very good reason to do so.
*#* Think of your own edge as a whole row of stones of your color. You already have a bunch of stones (and thus strength) there, and playing yet another one close by would be a waste. That stone is much better served elsewhere, where you don't have as much strength.
*# Bridges are the most efficient way to connect your own stones—not adjacent stones, trapezoids, crescents, or hammocks. If there are no opposing stones nearby, playing adjacent to an existing stone of your color is usually a mistake; if you want to do that, consider whether there's a better move a bridge away from that stone.
*#* If there are nearby opposing stones, things become more nuanced, but bridging to your existing stones is often still a good idea.
*# Usually, cut through an opponent's bridge if given the opportunity, and avoid letting it happen to you. In the pattern below, Red cuts through Blue's two stones a bridge apart. This is very good for Red and bad for Blue, because it's very hard for Blue to connect the two stones now.
*#: <hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 c1 R b1 1:b2" />
* This guide assumes you've read [http://www.mseymour.ca/hex_book/hexstrat.html ''Hex: A Strategy Guide''] or have equivalent knowledge.
** ''Hex: A Strategy Guide'' predates KataHex, but it's very well made, and 90% of the advice is still good.
* On smaller boards, tactics and raw calculation matter more. On larger boards, strategy matters more. (It's ultimately a spectrum, and it varies by skill level: for beginners, size 13 might be considered large; for strong players, tactics can matter a great deal on size 13.)
** On smaller boards, the exact board size affects strategy (some moves can be good on 11×11 but not 12×12). The larger the board, the less the exact board size matters. The general strategy for 19×19 and 20×20 should be nearly identical.
** On smaller boards, there is more space close to corners and edges. On larger boards, there is more "open" space (hexes not close to corners and edges). I intentionally said "open" space and not "center" space, because there is not much special about hexes near the center of the board, except that they are far away from corners and edges. But on large boards, there are many such areas.
** In particular, the larger the board size (starting at around size 13–15), the more important it is to know how to play in "open" areas. Many existing guides cover corner/edge strategy heavily—and correctly so, as corners/edges remain important up to very large sizes (25–30 or so). People read those guides and often know how to play the first 10–20 moves of a 17×17 game, but often not after that, when most of the game is still remaining!
** Strategy can also be ''local'' or ''global''. Local strategy tells you how to play within a small region of the board, and global strategy tells you how to play in relation to the whole board (or across multiple local regions). Both are important.
** The strategy of "open" areas is mostly local, and so the larger the board, the higher the relative importance of local strategy. Why?
*** On a large Hex board, the "middle" is relatively homogeneous—two separate open areas near the middle are basically the same in terms of how you should play them; it doesn't matter that one of the areas is 15 hexes away from an edge and the other one 18. Because of this homogeneity, the same basic principles will apply multiple times in the span of a game.
*** Knowing how to play across multiple local regions is very hard, and it's often different across games (especially for humans). It's better to master local play first, since similar local play principles will be relevant every game.
* How should a game of Hex (on larger boards) broadly feel like?
** While the win condition is binary (you either connect or you don't), the actual gameplay feels like a gradual accumulation of tiny advantages and disadvantages. Strong players often "know" they're winning in the middlegame, not because they see an actual way to connect, but because their opponent made some mistakes, however small, along the way. (You rarely know for sure who's winning until the end, and you're ultimately making an educated guess.)
** In particular, it's usually very hard to tell who's winning just by looking at a position. Contrast that with games like chess, where you can often tell by counting up material and broadly looking at positional factors. In Hex, it's much more tractable to tell how good your position is ''relative to a few moves ago.''
** Relatedly, there's (almost) no meaningful notion of "long-term strategy" in Hex. If anything, it can be harmful to create a long-term plan and stubbornly stick to it, because the moves you should play often differ greatly depending on your opponent's responses.
** Hex strategy is equally what to play and what ''not'' to play. The latter is frequently underemphasized. Each stone is worth a lot, and even one bad move can "undo" 10–20 good ones. The risk-reward balance strongly favors avoiding mistakes, rather than attempting risky but potentially "brilliant" moves.
* Now some points about how to learn. From what I observe, people often read strategy advice and then fail to follow it consistently. This is often either because of skepticism, or not realizing just how often particular pieces of advice should be followed (it's tempting to think that you've found the exception to the rule).
** It's healthy to be skeptical—there is a lot of ''bad'' strategy advice, and it would probably do more harm than good to implement every piece of advice you come across. So I'm not expecting readers to take everything in this guide at face value either (and many statements have nuances/exceptions, some of which I don't even realize should be explained to readers; it's hard to translate precise intuitions into words after all). Unfortunately, separating the signal from the noise is hard. If you're skeptical about a principle or how strongly it holds, try running it through KataHex yourself (if you have it installed), and try variations to understand the nuances, when the principle holds, and importantly ''why'' it holds. Sometimes people are skeptical of KataHex itself or think they can do better; I think there are good heuristic reasons why this thinking is misguided (the most simple one is that KataHex is far stronger than the strongest humans).

==General==

===Common mistake: playing near your own edge===

Playing near your own edge with no nearby stones is a bad idea in the opening and even middlegame. This is one of the most common mistakes among human players. You need a strong (often tactical) reason to do otherwise.
* When is playing near your own edge a good idea? There are a few cases, like when you're playing adjacent to existing stones, responding to a joseki, or pushing a ladder as the attacker, but all of these involve stones in the vicinity.
* This holds most strongly for 13×13 to 19×19, but applies more often than not to 11×11 too.
* Playing near your own edge in the opening or early middlegame should be a "last resort". I would recommend never doing so in the first 30-40 moves in 19×19 unless you have a very good reason.

===Common mistake: bad minimaxing===

Many intermediate and strong players are too eager to minimax. They play bad minimaxing moves (that KataHex thinks are inferior to directly connecting) more often than they fail to minimax. The issue is that minimaxing isn't always free; your opponent can often intrude for useful territory, and sometimes the territory your opponent gains is more useful than what you gain by minimaxing. It's a difficult skill to judge which player gains more from a minimax, but there are common patterns.
* Players frequently attempt to minimax by playing adjacent to their existing stone on the 5th or 6th row of their own edge. Playing two adjacent stones like this is not very efficient and typically a mistake.
* Blue's minimaxing attempt is unsuccessful; KataHex already thinks Red is 98% to win: https://hexworld.org/board/#15nc1,c2e11c13e10b12e9c10d8b9c7a8b6c6b7d7c8e8d9
* A common example; Red is 98% to win: https://hexworld.org/board/#13nc1,c2d10c11c10b11c9b8a10. Note that Red 7 at b8 is stronger than at a10, because b8 combines well with c2 and b5 on 13×13. On 15×15, KataHex prefers 7. a12 (the equivalent of a10 on 13×13) instead.
* A case where minimaxing is in fact the best move: https://hexworld.org/board/#13nc1,c2d10c9d11

===Common mistake: bad bridge peep===

A common scenario is the [[Peep#Bridge_peep | bridge peep]]. Blue should play at (*) as long as she thinks it's forcing enough for Red to defend his bridge.

However, even if Blue hasn't played (*) yet, Red should usually mentally place a blue stone there. Especially on larger boards, Blue will be able to play at (*) sometime in the future. This is important: even if (*) is empty, I like to think of this pattern as ''implying'' a blue stone at (*) (in the absence of tactics that somehow prevent Blue from playing there). Another way (useful for me, but the analogy might be nonsense to others) to think about it is that Blue's stone at A is well-placed enough relative to Red's bridge that she has some "potential energy" at (*).

<hexboard size="3x3" coords="hide" edges="hide" contents="R c1 b3 B A:a2 E *:c2" />

However, 80% of the time, if there's no blue stone at A, Blue should not intrude on either side of the bridge. Intruding is usually a mistake because it settles the [[question]] for Red. If Red knows he will have a stone at A or B because Blue already intruded in the other spot, Red can better plan his future moves because of that! There are exceptions; if Blue is confident that only one of the intrusions will ever be useful for her, then intruding is often strong.

<hexboard size="3x2" coords="hide" edges="hide" contents="R b1 a3 E A:a2 B:b2"/>

===Ladders===
* Don't worry about switchbacks, climbing, or ladder creation templates on 15x15 or larger. They can be theoretically interesting and make for good puzzles, but games are rarely close enough for these to matter. I personally don't think about them on any board size, but others may reasonably disagree with this approach.
* Q: When should the defender of a ladder yield? A: Usually, don't yield unless your opponent can cut through two of your stones, one above and one below the ladder. In that case, consider yielding.
** [https://hexworld.org/board/#13nc1,c2e9c10d8b9c11d10d12 Here] is an example. If Blue instead plays 8. d11, Red 9. e10 cuts through e9 and d11.
** In [https://playhex.org/games/66b8a125-bf02-42d2-8900-af3042a8ccf8 this game] from the [[Tournaments#Hex_Monthly | Hex Monthly 21]] ([https://hexworld.org/board/#14c1,c2d11k10k4b5f10g9i9i8g10h9h10j8j9b12c10l8k8l7k7l6k11j11h12m5k6l4m4l5l9k9h13i13l3k5i5j5j4i6h6f8h7i7f9g8f7e8e7d8d7b8c8b9c9a11b13d10c11c12,c13d12d13e11:rw HexWorld link]), the yielding move 60. c14 is winning, while the played 60. c13 (which allows Red d12, cutting through Blue's c13 and d11 stones) is losing.

===Other===

* Be aggressive; always be ready to capitalize on an opponent's mistake. Here are a few illustrative examples of this and some of the above concepts:
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2368410&nmove=8 this game], 6. c9 was a mistake (and an example of incorrect minimaxing); 7. a10 is the [[Openings_on_19_x_19#4-4_obtuse_corner | standard reply]]. White should have played 8. b8; j4 was a mistake. Black's 9. b8 is the "capitalizing" move; KataHex already says 99% win rate for Black. In this case, there were several good options on move 9. These alternatives are threatening enough that Black didn't have to play b8, but this won't always be true, like in the next example.
** [https://littlegolem.net/jsp/game/game.jsp?gid=2362913&nmove=26 Here] is another example (interestingly, White resigned in a winning position according to the bot, though the final position is highly tactical). KataHex actually thinks Black has only an 35% win rate after 26. m6. However, 27. d11 is a "minimaxing with adjacent stones" mistake, bringing Black's win rate down to 3%.
** [https://www.littlegolem.net/jsp/game/game.jsp?gid=2400947&nmove=6 Here] is an example where White's win rate increased consistently from 65% after move 6 to 98% after move 16. The aggressive bridging sequence in moves 8–16 is very strong because it basically neutralizes Black's stones on d16 and d17; it is a big mistake for Black to ignore the threat here. Move 11 is also an example of a bad bridge peep.

* Don't play a move that makes your opponent's existing stones unnecessarily well-placed relative to your new stone. This is covered briefly in a couple places in the [[Openings_on_19_x_19#General_principles | 19×19 opening guide]]. Here are a couple more examples.
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2369954&nmove=26 this game] ([https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6 HexWorld link]), what should Black play on move 27? White is threatening to connect 26 back to the 16–24 group if Black isn't careful. Black's obvious options are to play at c8 and c9. Which option is better?
** A tempting answer is c8 because it appears to connect more closely to the top. However, note that White's stone on move 14 functions exactly like an opening a3 stone. For example, it can escape 2nd but not 3rd row ladders. If Black plays c8, White can play the [[A3 escape trick]], as shown in [https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6c8e7d8e8d9e9 this continuation]. This sequence allows White to "make the most" of stone 14, and Black shouldn't allow it. Better for Black is 27. c9, which doesn't allow White to carry out a 3rd-to-5th row switchback. Indeed, KataHex evaluates c9 as 97% to win for Black and c8 as only 30%.
** Consider [https://hexworld.org/board/#13n,c2d10c11c10b11e6b10d8l4l3c8d5b5d6h7h6j5i6k7 this position] (which is based on [https://www.littlegolem.net/jsp/game/game.jsp?gid=2369782 this game]). Where should White play in the lower-right acute corner? Black had just played at the "3-7" point from White's perspective; as mentioned in the 19×19 opening guide, a strong response to 3-7 is 4-4 (at j10) instead of the usual 5-4 (at j9). A player who is short on time can play j10 without thinking too hard. As it turns out, KataHex thinks j10 is the best move, evaluating it as 91% for White, whereas j9 is only around 50%.

* Sometimes, you know you have a completely winning position after your opponent blunders very early in the game. Your goal is to preserve that win until the end of the game. Don't overextend yourself and try to win too "quickly", possibly making suboptimal moves in the process. Even when KataHex plays itself with a significant handicap (like playing without swap), a significant fraction of the board seems necessary to carry out the win, assuming strong defense. I prefer taking my time to "fill the board" by focusing on playing moves at least as good as my opponent's, instead of worrying about connecting in as few moves as possible.

==Local patterns==

An especially effective way to improve at Hex is to learn and gain intuition about lots of small local patterns. These patterns are small enough that they occur basically everywhere when you play Hex. Knowing the patterns can help you avoid blunders (for example, avoiding adjacent or bridge ''allow-cuts'', as explained below) and make good moves when it's otherwise unclear what to play.

===Cuts===
<hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 c1 R b1 1:b2" />
<hexboard size="1x3" float="inline" edges="none" coords="none" contents="B a1 c1 R 1:b1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" visible="-(a1 b1 d1 a3 c3 d3)" contents="B a2 d2 R c1 1:b3 S b2 c2 E *:(b2 c2)" />
<hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a3 c1)" contents="B a1 c3 R 1:b2 S b1 a2 c2 b3" />
Consider the four patterns above. Let's call the patterns '''adjacent cut''', '''near cut''', '''short bridge cut''' (or just '''bridge cut'''), and '''long bridge cut''' respectively. In some sense, Red is "cutting" through two of Blue's stones in each pattern, so Red is the '''attacker''' and Blue is the '''defender'''. We can also use '''cut''' as an attacking verb and '''allow-cut''' as a defending verb. For example, the adjacent cut pattern is favorable for the attacker (Red above), so we can say that it's a good idea to ''adjacent-cut'' through opponent stones, and a bad idea to ''adjacent allow-cut'' (meaning it's a bad idea to play a move that allows your opponent to adjacent-cut through two of your stones).

Why is it strong to adjacent-cut (and equivalently, why is it a mistake to adjacent allow-cut)? The most intuitive reason is that if Blue allow-cuts, it takes 3 extra moves for her to connect her two stones, as shown below. In a sense, this makes it unlikely for both stones to be concurrently useful in the final connection — if one of the stones proves useful, chances are the other one won't be.
<hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 c3)" contents="B a2 c1 1:a3 2:b3 3:c2 R b1 b2" />

The (short) bridge cut is strong for the attacker (Red above) because he captures the two hexes marked (*). If Blue allow-cuts, often the damage has already been done even if Red doesn't actually cut through immediately on the next turn. (However, note that this isn't always true; some tactical situations require Red to cut through immediately instead of playing elsewhere.)
* There are situations where Blue ''should'' allow-cut. If Red's cutting move comes with concessions, like the move is on his second or third row (and especially if Red already has a stone near his own edge, so cutting through would be overplaying that edge), the pattern is much less favorable to Red.

The near cut and long bridge cut are also favorable for the attacker, but much less clearly so — there are many more "exceptions" where cutting is bad and allow-cutting is fine. On the other hand, the adjacent and short bridge cuts are quite strong, and cutting through is usually the best move (unless doing so involves playing a stone that's too close to your own edge, or another major concession).

Note that allow-cuts can occur not only ''proactively'' before the attacker plays the cutting move, but also ''retroactively'' after the "cutting" move is played (of course, it's not really a "cutting" move until after the defender allow-cuts). In other words, Blue 1 is considered an allow-cut in both of these examples (proactive on the left and retroactive on the right), and they are equally bad for Blue assuming that Red intends to cut in the proactive case:
<hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 1:c1 R b1" />
<hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 1:c1 R b1 b2" />

An important way of pruning moves and avoiding blunders is making sure you don't play moves that adjacent or bridge allow-cut, either proactively or retroactively, unless you have a good reason to do so. In particular, the retroactive case can be easy to overlook if you're not used to thinking in this way.

===Small patterns===
<hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="B b2 R b1 1:a2" />
<hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a3 c2 c3)" contents="B b3 R c1 1:a2" />
<hexboard size="4x2" float="inline" edges="none" coords="none" visible="-(a1 b4)" contents="B a4 R 1:b1" />

In each of these patterns, we will discuss the strength of Red's move 1 ''in a vacuum'', meaning in the absence of nearby stones, like near the middle of an empty board on 19×19 or even larger boards. It is most useful to analyze each pattern without the influence of nearby stones or edges (which can be thought of as virtual rows of stones).
* Pattern 1: It's typically an inaccuracy for Blue to play adjacent to a red stone in a vacuum. Red's best response is usually playing adjacent to both stones, as in Red 1.
* Pattern 2: Blue plays a bridge away from Red's stone. This is often a good move, and Red 1 a bridge away from both stones is also good.
* Pattern 3: Red 1 is a classic block. This is often fine for Red, and Blue doesn't have an obvious best local response. Blue could tenuki here.

===Capture patterns===

Know the most common [[captured cell]] patterns. It really reduces the number of moves you have to consider.

The bridge cut is one of the most important capture patterns.

* A typical example can be found in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2233736&nmove=10 this game]. After 10. g11, Black would have the easiest time with 11. h9, with 95% win rate, instead of the played k8 with 74% win rate. After k8 was played, one of White's best moves would've been h9 to prevent Black from playing the same. KataHex does think Black has other reasonable options on move 11, but they mostly appear to just delay h9.

Here is another common capture pattern. The other patterns on [[captured cell]] are useless because they have too many Red stones adjacent to each other, which is too inefficient to occur in normal play.

<hexboard size="3x3" coords="hide" edges="hide" contents="R c1 c3 b3 B a2 S b2 c2 E *:(b2 c2)" />

This ''one-sided'' capture pattern is very common. I like to mentally replace A with a blue stone and (*) with a red stone (even though Blue won't always achieve this), because Blue should never play at (*). [[Edge template IV3a]] is useful to know here.
<hexboard size="3x2" coords="hide" edges="hide" visible="-a1" contents="R b1 a3 b3 S a2 b2 E *:b2 A:a2" />

The trapezoid and crescent also one-sided-capture the hexes at (*). Anecdotally, this fact is more useful on smaller boards like 11×11, but it's less common overall than the previous capture patterns.
<hexboard size="3x3" float="inline" coords="none" edges="none" visible="-a1" contents="R b1 a3 c3 c1 S a2--c2 b3 E *:(c2 b3)" />
<hexboard size="3x3" float="inline" coords="none" edges="none" visible="-a1" contents="R a2 b1 c1 c3 S b2 c2 a3 b3 E *:c2" />

==How to think about the opening and middlegame==

Some of these points sound like hyperbole, but they are a fairly accurate representation of how I think about Hex, especially on larger boards. There are exceptions, but exceptions are rarer than one might think.

* In the first half of the game, focus all of your attention into playing "efficient" moves. These are moves that work well with existing stones and board edges. Excellent intuition of efficient patterns is enough to get strong middlegame positions against even top players (as of 2024).
* Don't worry about tactics until the late middlegame. It's a waste of time to analyze sequences >10 moves deep. In the opening, the best move you can find with only intuition is no worse than the best move you can find with calculation.
** Why? Compared to bots, humans are very bad at calculation. Your intuition (which you learn directly from KataHex) is more useful than hypothetical hours or days of error-prone human analysis.
* Use your knowledge of efficient patterns to plan ahead. This is where you want to do shallow analysis, typically 1-5 moves deep. A good move does as much as possible:
** allows you to play efficient moves later
** prevents your opponent from playing an efficient move outright
** allows your opponent an efficient move, but only with concessions (say, a locally efficient but globally inefficient move, or a move that's efficient with relative to one stone but inefficient relative to another)

These points require a lot of elaboration. What do I mean by "efficient"? (Not to be confused with [[efficiency]].) There are two broad notions, '''global''' and '''local''' efficiency.
* Globally efficient stone: Broadly, any move that isn't too close to your own edge. Moves that are strong in the absence of nearby stones (like on an empty board). Corner moves are very efficient. On 19×19, moves near your opponent's 5th row are very efficient.
* Locally efficient stone: In the presence of nearby stones, some configurations are broadly "good" or "bad" for one player. They aren't ''always'' good or bad because of tactics, but because humans can't calculate everything, it's extremely useful to have decent heuristics that work most of the time.

In a sense, you want to play the move that maximizes the "sum" of local and global efficiency. This isn't an exact science; use your intuition to make an educated guess.
* The best move locally is rarely the best move globally. Hex is a game of concessions and tradeoffs. There is no "free lunch," unless you count concepts like inferiority/domination.
* Conversely, if it looks like your opponent is beating you on one side of the board, there are two possibilities:
*# You played suboptimal moves in that region.
*# Your opponent overplayed that region, which necessarily means they made concessions elsewhere (usually the opposite side of the board). This means your opponent ''underplayed'' the other region.
* If option 1 is true, you are probably losing anyways, so assume (or hope) option 2 is true. If you opponent overplays region A and underplays region B, it's typically a good idea to focus your play in region B. Why? If you keep playing in region A, you're potentially helping your opponent out of their mistake, by making your opponent less over-connected in region A. For example, your opponent could have a minimaxing move that makes them much stronger in region B, at the "concession" of making them weaker in region A — but this isn't really a concession because your opponent had strength to spare in region A.

The discussion so far has been highly abstract. What does efficient play look like in practice? As explained earlier, it's strong to adjacent-cut and bridge-cut through your opponent's stones. The small patterns in the "Capture patterns" section above are also locally efficient for the capturing player. Of course, just as offense and defense are equivalent in Hex, so are playing locally efficient moves and inducing your opponent to play locally ''inefficient'' moves.

In the following examples, P1 and P2 are efficient patterns for Red relative the corresponding edges, while P3 is inefficient. P1 is common when Red opens with a first column opening. P2 requires elaboration: normally Red 1 is a weak move because it's too close to Red's own edge. However, when Red must play on his first 3 rows (because of very limited space), Red 1 or its mirror image at (*) is usually the best, because it prevents Blue from playing there and forming the equivalent of P1 herself. P2 also occurs in theoretical contexts, like [[Template Va#If Blue moves at y:]]. In P3, Red 1 is a tempting (because of [[edge template III1b]]), common, but inefficient move, because Blue's response at one of (*) is usually strong. If Red must play near his own edge, he should instead play as in P2 above.
P1: <hexboard size="3x2" float="inline" edges="left" coords="none" contents="R a2 1:b3" />
P2: <hexboard size="2x6" float="inline" edges="bottom" coords="none" contents="B c2 R 1:e1 E *:b1" />
P3: <hexboard size="3x5" float="inline" edges="bottom" coords="none" contents="B c3 R 1:d1 E *:(b2 e2)" />

===Making inferences: An extended example===

Use the patterns and intuitions that you know, to reason and make inferences in unfamiliar positions. You might not always find the best move, but you can often prune bad moves. Here is an extended example, involving several closely related patterns that are useful to know in isolation.
# ''Pattern:'' If you play the 4-4 3-3 obtuse corner joseki, it is important to know that Blue 6 [https://hexworld.org/board/#19nc1,d16c17d17c18d18e14c15 here] is weak.
#* This is because Red 7 undermines Blue 6 and also reduces Blue 2/4 from a 3rd row ladder escape to a 2nd row escape. Also, Blue's potential intrusion at c16 doesn't gain her anything.
#* Instead, [https://hexworld.org/board/#19nc1,d16c17d17c18d18d15 here] is a good local response for Blue. This gives Blue a 5th row ladder escape while blocking Red 1.
#* On the other hand, if Blue only has a 2nd row escape and doesn't have the "potential energy" to get a 3rd row escape, you might guess that Blue 6 above becomes a good local move, because Red 7 loses one of its main strengths. Such a guess would be correct: in [https://hexworld.org/board/#19nc1,d16c17p15d17e16g16e17f18d18e14c15 this position], Blue 10 is a strong move. On the other hand, Blue 10 at d15 would be weak here, as it's quite wasteful, only converting a 2nd row escape into a 3rd row escape.
# ''Pattern:'' If Red tenukis on move 3, then Blue 4 is locally strong in [https://hexworld.org/board/#19nc1,d16e14:pf15 this position]. In particular, Blue 4 is much stronger than bridging directly towards Blue's own edge with g13, because the former is a strong blocking move that reduces the effectiveness of Red 1. This strength is in spite of Red's intrusion on move 5. (I first saw this move in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2090091&nmove=16 this bot game] from [https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=739 this thread on LittleGolem].)
#* Variations of this pattern appear frequently in obtuse corner joseki. If Red started with the 5-5 obtuse corner, then the equivalent Blue 4 is [https://hexworld.org/board/#19nc1,e15f13:pg14 also] strong.
#* Another simple example is Red 5 in [https://hexworld.org/board/#19nc1,d16d15f14:pe13 this position].
# ''Inference:'' Let us consider [https://hexworld.org/board/#19nc1,d16c17d17c18d18:p this position] again. Suppose Blue tenukis on move 6. What is a good local response for Red on move 7? There is more than one right answer, but you just need to find one move that isn't a blunder.
#* ''Hint:'' Use the two patterns above to come up with an answer.
#* ''Hint:'' Consider that Blue has a 3rd row ladder escape, so Blue d15 is locally strong but Blue e14 is weak. This is the case for future blue stones in this local region, even though Blue already played elsewhere on move 6. Therefore, Red should perhaps try to play a move that makes Blue d15 less attractive, even if it comes at the concession of making Blue e14 more attractive (since the latter is ''a priori'' a weak move, such a concession gives up less than it otherwise might).
#* ''Answer:'' Red 7 [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13 here] is a good local move.
#** Why? Ignoring stones 2-5, the presence of Red 1 and 7 makes Blue e14 more attractive. Red's 4-4 stone, much like the obtuse corner opening, tends to shift Blue's efficient stone locations up one row. With that in mind, Red's move 7 is a bit like Red e14 on an empty board, where Blue e15 is [https://hexworld.org/board/#19nc1,e14e15 the best response]. However, stones 2-5 make Blue e14 a bad move, despite Red 7 making it slightly less bad (this is the concession-but-not-really-a-concession that Red intended).
#** Let's now consider Blue d15. Normally a decent local move after stones 1-5, it becomes quite weak after Red 7, because of [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13d15f14 this sequence]. This is a typical example of the second pattern in this extended example. (Actually, if Blue had two moves in a row in this region, Blue 8 wouldn't be so bad because she could follow up with f14, where Red wants to play move 9. This is a tactical situation where it's bad for Red to tenuki.)
#** If we wanted to be complete, Blue actually has another option for move 8, f14 first which threatens d15: [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13f14d15d14e14 here]. This is an example of the "skew cut" from above. Red should respond with d15 himself, or else Blue can play d15. This is probably Blue's best local response.
#** Every move makes a concession, but some concessions are more effective than others. As alluded to in the hint, Red 7 weakens a strong Blue reply (d15) while strengthening a weak Blue reply (e14). Red's goal is to minimize the strength of Blue's ''best'' response (rather than the strength of Blue's average response), and Red 7 at e13 accomplishes that beautifully.

Part of getting stronger at Hex (as with other games) is being able to compress knowledge and patterns effectively. If you understand this extended example, you can infer the best move in many similar but novel situations, without memorizing each one individually. If you forget which move is exactly the best, you can recover the right move with high probability by just reasoning about related patterns that you do know are strong or weak. The "A3 escape trick" example earlier in this article is another instance of this concept.

On larger boards and particularly near the opening or middlegame, when you have a lot of time to think about a position, it may be worth spending a good chunk of it making inferences and scoping out moves that are either good or to be avoided by each side, rather than spending most of your time calculating tons of lines and variations.
* In fact, once you've scoped out that "moves A and B are good for Red, while moves X and Y are bad for Red" and similarly for Blue, that makes the process of analyzing and pruning lines much more efficient.

==Joseki==

* The 4-3 corner move is almost always a mistake compared to the 5-4, even on 11×11. If your opponent plays 4-3, 90% of the time the best local reply is 3-3. If 4-3 is played, here are some common sequences:
: <hexboard size="5x7" float="inline" coords="hide" edges="bottom right" contents="R 1:e2 B 2:e3 R 3:d3"
/> <hexboard size="5x7" float="inline" coords="hide" edges="bottom right" contents="R 1:e2 B 2:e3 R 3:f2 B 4:f3 R 5:d3 E a:b4 b:c5"
/>
: In the second joseki, Blue should play either (a) or (b), or she can defer the question until later. On larger boards with no nearby stones, (b) is often better.
* If your opponent plays the 5-4 joseki, a reasonable choice is to always play the 4-4 "high intrusion." KataHex prefers the 4-4 response by far, especially on 13×13 and larger. It's easily the "safest" choice, and many strong players have recently shown a clear preference for 4-4.
* In this often-played joseki, Red 7 is a mistake because of the territory Blue gets after move 8:
: <hexboard size="6x10" float="inline" edges="bottom right" coords="none" contents="R 1:g2 B 2:g3 R 3:h2 B 4:h4 R 5:i3 B 6:i4 R 7:f3 B 8:e5 R 9:d4 B 10:c6 R 11:b5 B 12:c4 R 13:c5" />
:* If the obtuse corner on Red's same edge is empty, a more principled response for Red is to play in that corner, like this:
:: <hexboard size="6x14" float="inline" edges="bottom left right" coords="none" contents="R 1:k2 B 2:k3 R 3:l2 B 4:l4 R 5:d3" />
:* If the obtuse corner is already occupied by Blue, then Red should still respond in the obtuse corner. One reasonable option (among others) is this:
:: <hexboard size="6x14" float="inline" edges="bottom left right" coords="none" contents="B d3 R 1:k2 B 2:k3 R 3:l2 B 4:l4 R 5:e3" />
* See more discussion of acute corner joseki at [[Openings_on_19_x_19#Acute_corner_theory]]. For the obtuse corner, see [[Openings_on_19_x_19#Obtuse_corner_theory]] which covers responses to the 4-4 and 5-5 obtuse corner in great detail. Most of this is applicable to 13×13 and larger, and not just 19×19.

==Specific tactics==
===Threatening a bridge cut===
One of my favorite tactics is threatening a bridge cut, where the opponent needs to play too close to their own edge to defend:
<hexboard size="8x8" float="inline" edges="bottom left" coords="hide" contents="B *:d5 g2 R 1:f4 B 2:e3" />
Red 1 is a strong move, because it forces Blue to respond at 2 (or else Red will play the bridge cut there), but Blue 2 is normally overplaying on the left side when combined with Blue's stone at (*). Be on the lookout for this when your opponent plays the 4-4 and 7-7 obtuse corner, or 5-5 and 8-8.

This tactic can work well in other configurations, like [https://hexworld.org/board/#19c1,:pg8:pg11h9 this] and [https://hexworld.org/board/#19c1,:pe9:ph9g8 this]. However, be careful when using this tactic! When Blue isn't actually making a concession by bridging her two stones, like when the pattern occurs further from her own edge, it can be a blunder for Red to threaten a bridge cut. [https://hexworld.org/board/#19nc1,:pg8:pj5i7h6 Here] is an example where Red 5 is a blunder in the absence of nearby stones.

[https://www.littlegolem.net/jsp/game/game.jsp?gid=2436000&nmove=12 Here] is an example from a real game. Black's best move is 13. h5 (in the actual game, Black doesn't play this until move 19 but is still arguably rewarded for doing so). One might think that h5 is too strong near the top edge because of c2 and b5, but this is not the case because the white stone on c5 partially neutralizes b5.

[[category:strategy]]

User:Hexanna

2025-12-14T22:09:33Z

Hexanna: finish up 2-stone discussion

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.35-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki, is close to optimal.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. All other moves are almost equally good. [Playing Δ1 from S is a 0.05-stone mistake, playing Δ3 away is a 0.1-stone mistake, and everything else is close to optimal.]
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising and doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(e) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
:** (d) and (e): Most simply, these Red replies are weak because they're adjacent to Red's existing stone, and Blue's existing stone isn't sufficiently close enough to make the second red stone useful. KataHex says this is a 0.45-stone mistake.
:** (f): This is a long bridge allow-cut. According to our mental model, this should be 0.3-stone mistake — 0.2 from the tenuki baseline, and 0.1 from the allow-cut. KataHex agrees.
:** (g): This is a (short) bridge allow-cut. KataHex says it's roughly a 0.6-stone blunder.
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-14T21:45:24Z

Hexanna: more

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. (Technically, many KataHex versions don't work well with pass moves, so instead, we estimate the value of the worst move on an empty board — it's pretty close to zero — and use that instead.) Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.35-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki, is close to optimal.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. All other moves are almost equally good. [Playing Δ1 from S is a 0.05-stone mistake, playing Δ3 away is a 0.1-stone mistake, and everything else is close to optimal.]
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising but doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:* If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(e) and tenuki.
: <hexboard size="4x6" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4) f:f1 g:e3" />
:* Red is already ahead by 0.1 stones because Blue played a bridge away from Red.
:* Red's optimal local response is (a), preserving the 0.1-stone advantage. This is an important pattern, and we will say much more about it later. [''Reminder to self:'' Talk about this pattern but extended, the bridge intrusion, the retroactive bridge allow-cut and its role in pruning relative to the bridge intrusion.]
:*: <hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a2 b3 c3)" contents="R a3 B c2 B b1 S c1 b2" />
:* ''Quiz:'' how much worse than (a) is each of the other marked moves (including tenuki)?
:* ''Answers:''
:** Tenuki: By symmetry, if Red tenukis, Blue can play at (a), turning a 0.1-stone advantage into a 0.1-stone disadvantage, so tenuki is a 0.2-stone mistake. KataHex agrees, and just like in the adjacent case, this pattern is tactical.
:** (b): This is a near allow-cut, and after Blue responds at the adjacent (c), the resulting pattern is congruent to case (e) in the previous discussion of the adjacent 2-stone pattern (where KataHex said it was a 0.25-stone mistake). However, unlike the previous case, our baseline is higher: Red is ahead by 0.1 with the optimal response, not 0.05. It's worse (by 0.05) to start from a 0.05-stone higher baseline and reach the same result, so this is actually a 0.3-stone mistake.
:** (c): This is congruent to case (c) in the adjacent 2-stone pattern (which was a 0.4-stone mistake), but we're starting from a 0.05 higher baseline, so this is a 0.45-stone mistake.
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-14T20:53:45Z

Hexanna: more

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.25-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki, is close to optimal.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. All other moves are almost equally good. [Playing Δ1 from S is a 0.05-stone mistake, playing Δ3 away is a 0.1-stone mistake, and everything else is close to optimal.]
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent opposing stone, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable predictions. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) it as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising but doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, KataHex evals are inherently noisy, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)
:** If you understand the above, you only have to memorize that (a) is the optimal reply, because you can reconstruct nearly all of the rest from first principles and fuzzy reasoning.
* Let's consider the other important two-stone pattern, a stone of your color and an opposing stone a bridge away. Suppose it's Red's turn. We can do a similar exercise and evaluate (a)–(e) and tenuki.
: <hexboard size="4x4" float="inline" edges="none" coords="none" contents="R b3 B d2 E a:(c1 c4) b:(d1 d3) c:(c2 c3) d:(b2 b4) e:(a3 a4)" />
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-14T20:37:20Z

Hexanna: more local pattern stuff, "fuzzy reasoning"

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns===

[''prerequisites: cuts, maybe "pattern 1" and other patterns in that section — consider reordering'']

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.25-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki, is close to optimal.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. All other moves are almost equally good. [Playing Δ1 from S is a 0.05-stone mistake, playing Δ3 away is a 0.1-stone mistake, and everything else is close to optimal.]
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent stone of the opponent's color, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to make reasonable guesses. Analyzing real games often involves similar kinds of guessing.
:* ''Answers:''
:** Tenuki: The simplest response to analyze. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, implying that tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:** (b): This is an adjacent allow-cut. We haven't quantified these precisely, but you should expect (b) to be a blunder. KataHex says it's a 0.5-stone mistake, and Blue's best reply is cutting through at the adjacent (a).
:** (c): This is a mistake because Blue can respond at the non-adjacent (a), rendering (-) below relatively useless. KataHex quantifies (c) it as a 0.4-stone mistake.
:**: <hexboard size="3x2" float="inline" edges="none" coords="none" visible="-(b1 b3)" contents="R -:a2 B b2 R a1 B a3" />
:** (d): Blue can respond at either (a) and create a pattern congruent to the one when Red plays (c), so this is also a 0.4-stone mistake.
:** (e): The start of a [[bottleneck]], but more importantly, it's a near allow-cut. Applying our mental model: a near allow-cut is a 0.1-stone mistake, but given that Red didn't play (a), we should think of this on top of a tenuki baseline, which adds another 0.1 (total 0.2). KataHex actually says it's a 0.25-stone mistake. This kind of deviation is unsurprising but doesn't invalidate the mental model — our reasoning was a bit handwavy, Blue has alternatives to the near-cut, and we probably didn't account for all nonlinearities.
:** (f): This is a retroactive near allow-cut. Similar to (e), it's a 0.25-stone mistake.
:** (g): This and similar Red moves further away from the blue stone are harder to reason about. KataHex thinks these moves are generally 0.15- to 0.2-stone mistakes; working backwards, you could decompose that into a 0.1 tenuki baseline, and attribute the remaining portion to the "non-tactical" component of the move's strength. Also, it should be intuitive that the further away the Red move, the more it "looks like" tenuki, where the non-tactical component tends to zero in the limit. (Again: while this is handwavy, the general approach of "fuzzy reasoning" we used here — taking the heuristics you know to imply conclusions that aren't necessarily "correct" in some theoretical sense, but that are much better than blind guessing — is a powerful tool when thinking about real games.)

* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-14T19:29:22Z

Hexanna: more local pattern stuff

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns===

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.25-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki, is close to optimal.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. All other moves are almost equally good. [Playing Δ1 from S is a 0.05-stone mistake, playing Δ3 away is a 0.1-stone mistake, and everything else is close to optimal.]
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent stone of the opponent's color, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which was a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* Red's optimal local response is (a), preserving the 0.05-stone advantage. This is a very important local pattern! (It's also our first nonlinearity. While playing adjacent to your own stone ''in a vacuum'' is a mistake since you're overplaying that region — the second stone is partially redundant with the first — the presence of the opposing stone makes that no longer true. As we'll see later, an important topic is learning how opposing stones influence your own.)
:*: <hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="R a2 B b2 R b1" />
:* ''Pop quiz:'' how much worse than (a) is each of the other marked moves (including tenuki), in fractions of a stone, assuming no other major nonlinearities? This is a tricky exercise, but if you understand the mental model well and know about cuts, you already know enough to do reasonably well. Analyzing real games (under this model) often involves similar kinds of reasoning.
:* ''Answers:''
:** The simplest response to analyze is tenuki. If Red tenukis instead of playing (a), then by symmetry Blue could play (a) herself! That flips a 0.05-stone advantage into a 0.05-stone disadvantage, so tenuki is a 0.1-stone mistake. Indeed, KataHex agrees when you check the evals. In a way, the strength of (a) is why Blue's earlier choice to play adjacent to the Red stone is locally slightly inaccurate. And because tenuki is suboptimal, the local pattern is slightly tactical; this contributes to Hex games feeling like lots of local fights rather than a "cold" game.
:**

* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-14T17:20:11Z

Hexanna: mental model; start talking about 2-stone patterns

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns===

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.
** One mental model we'll use: you can quantify how much a position/pattern/move favors one side in terms of ''fractions of a stone''. The basic idea is that [[passing]] is like giving up "1 stone" (and so it's worth 0 stones), the best move is worth 1 stone, a fair swap move is worth 0.5 stones, and moves can also be worth >0.5 or <0.5 stones. The same idea can be extended to patterns. The crux of the mental model is that empirically, the [https://en.wikipedia.org/wiki/Superposition_principle superposition principle] ''approximately'' applies, especially when multiple patterns are sufficiently far apart and independent. That is, if Red made a 0.2-stone mistake somewhere and Blue made a 0.1-stone mistake elsewhere, in some sense, your best guess is that Blue is ahead by 0.1 stones.
*** This is a heuristic mental model — while I think it's ''very'' useful, it's not the ground truth. (Of course, every position is theoretically either winning or losing, but that's not a useful lens strategically.) In general, Hex is "nonlinear", and nearby patterns can absolutely interact. However, empirically it behaves well and is ''locally'' linear, which is usually what matters strategically. Of course, if you subscribe to this model, part of Hex strategy is also thinking about the nonlinearities.
*** For the mathematically inclined: to evaluate a move or pattern, we consider KataHex win rates in logistic Elo space, and then we apply linear interpolation of the candidate move compared to the "optimal" and pass moves. Logistic Elo is the correct space to use because it's proportional to log odds, and adding Elo numbers is like multiplying odds, which is the correct way to apply updates in a Bayesian sense.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.25-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki, is close to optimal.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover soon. All other moves are almost equally good. [Playing Δ1 from S is a 0.05-stone mistake, playing Δ3 away is a 0.1-stone mistake, and everything else is close to optimal.]
* Let's move on to some important two-stone patterns. Consider a stone of your color and an adjacent stone of the opponent's color, like in the diagram below. Suppose it's Red's turn. Locally, how far from optimal is each marked Red response (a)–(g) below (where some responses have the same letter due to symmetry)? How far from optimal is it to tenuki?
: <hexboard size="5x5" float="inline" edges="none" coords="none" contents="R c3 B d3 E a:(d2 c4) b:(e2 d4) c:(c2 b4) d:b3 e:(e1 c5) f:e3 g:(b2 a4)" />
:* Red is already "ahead" by 0.05 stones because Blue played adjacent to Red (which is a 0.05-stone mistake). This is our baseline for evaluating Red's move.
:* ...

* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-14T15:03:14Z

Hexanna:

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

* terminology: inaccuracy < mistake < blunder
* evals are noisy, taking the highest one in non-tactical situations can be biased
* much later, talk about strength on an edge and how it behaves with existing stones

===Local patterns===

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are near and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.25-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki, is close to optimal.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover later. All other moves are almost equally good. [Playing Δ1 from S is a 0.05-stone mistake, playing Δ3 away is a 0.1-stone mistake, and everything else is close to optimal.]
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-14T01:45:56Z

Hexanna:

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

terminology: inaccuracy < mistake < blunder

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. A few points worth mentioning first:
* We'll sometimes use quantitative language, such as "fractions" of a stone. That doesn't mean you need to memorize the numbers or do a bunch of math while playing Hex. Numbers are just less ambiguous than words.
* Hex strategy is largely about making the right tradeoffs and avoiding blunders, not finding "the best move." Real games are messy. There's local/global considerations and the possibility of overlapping patterns or regions. For each pattern, try to learn not only the "optimal" move(s), but also roughly how bad the suboptimal moves are, so you can properly weigh moves that look strong in one dimension but weak in another.
* You won't remember everything here, and this guide isn't exhaustive. Your goal is to build a parsimonious mental model of Hex: what good/bad Hex looks like, and ''why'', so that your intuition generalizes well to new positions.

Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are adjacent and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.25-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki, is close to optimal.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover later. All other moves are almost equally good. [Playing Δ1 from S is a 0.05-stone mistake, playing Δ3 away is a 0.1-stone mistake, and everything else is close to optimal.]
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-13T22:38:06Z

Hexanna: clarification

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

terminology: inaccuracy < mistake < blunder

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intuitively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are adjacent and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.25-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki, is close to a 0-stone mistake.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy because your opponent can respond adjacent or a bridge away (respectively) from both stones, as we'll cover later. All other moves are almost equally good. [Playing Δ1 from S is a 0.05-stone mistake, playing Δ3 away is a 0.1-stone mistake, and everything else is close to a 0-stone mistake.]
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-13T22:32:59Z

Hexanna: simplest local patterns

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

terminology: inaccuracy < mistake < blunder

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* The simplest open patterns are a single stone of your color, and a single stone of the opponent's color. They aren't as interesting as other patterns, but we'll discuss them for completeness.
** Given a stone S of your own color in a vacuum, where should you play relative to it? Most choices, like playing a bridge or a "classic block" or much further away from S, are nearly equally good. One major exception is playing adjacent to S, which is a mistake (intutively, it's wasteful to play too many of your own stones close together). Playing two adjacent steps or two bridge-moves away is an inaccuracy, because those moves are adjacent and long bridge allow-cuts, respectively. [''Author's note:'' Using Δ-notation makes this much cleaner, but I'm not sure whether to introduce it in the actual article. If we apply some averaging and heuristic "noise reduction" methods to KataHex evals, playing Δ1 from S is a roughly 0.25-stone mistake in a vacuum, playing Δ4 or Δ12 from S is a 0.1-stone mistake because of the allow-cuts, and everything else, like Δ3/Δ7/Δ9/Δ13/Δ16 or tenuki is close to a 0-stone mistake.]
*** ''Caution:'' It's easy to draw incorrect conclusions from this if you're not careful. Keep in mind this is the strength of a stone relative to S ''without any nearby stones'', not even the edges/corners (which can be thought of as virtual stones). This never happens on small boards, and must be balanced with many other factors on larger boards.
** Given a stone S of the opponent's color in a vacuum, where should you play relative to it? Playing adjacent or a bridge away from S is a slight inaccuracy, and all other moves are nearly equally good. [Playing Δ1 from S is a 0.05-stone mistake, and playing Δ3 away is a 0.1-stone mistake.]
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

Claude Berge's puzzles

2025-12-13T19:32:27Z

Hexanna: fix puzzle 4 link (l10 instead of m10)

These are a few puzzles designed by [[Claude Berge]]. Their respective solutions can be found [[Solutions to Claude Berge's puzzles|here]].

== Puzzles ==
=== Puzzle 1 ===
[[Blue]] to move and win.
<hex>R5 C5 Q1
Ha1
Vb2 Hd2 Ve2
Vc3 Hd3 Ve3
Hc4
Va5 Hb5
</hex>
Try it on [https://hexworld.org/board/#5c1,a1:sc3d2e2d3e3c4b2b5a5 HexWorld].

=== Puzzle 2 ===
Blue to move and win.
<hex>R5 C5 Q1
Va2
Ha3 Vb3 Hc3
Vb4
Ha5
</hex>
Try it on [https://hexworld.org/board/#5c1,c3:sb3a3a2a5b4 HexWorld].

=== Puzzle 3 ===
Red to move and win.
<hexboard size="14x14"
coords="show"
contents="B d2 B f2 R b3 B d3 R g3 B i3 R a4 B b4 B e4 B f4 R g4 B j4 R a5 B c5 R g5 B l5 B a6 B b6 R c6 B d6 B j6 R c7 B e7 B h7 R c8 B f8 R h8 R j8 R c9 R c10 R c11 R c12 R c13"
/>
Try it on [https://hexworld.org/board/#14c1,b4:sb3f2g3d3a4i3g4b4a5e4g5f4c6j4c7c5c8l5h8a6j8b6c9d6c10j6c11e7c12h7c13f8 HexWorld].

=== Puzzle 4 ===
Red to move and win.

<hexboard size="14x14"
coords="show"
contents="B k1 R g2 B h2 R l2 R g3 B i3 B k3 R l3 R g4 B i4 R l4 R g5 B j5 R l5 R g6 B h6 B k6 R l6 R g7 B i7 R l7 R g8 B j8 R l8 R g9 B h9 B k9 R l9 R g10 B i10 R l10 R g11 B j11 R m11 B g12 B h12 R i12 B k12 R n12 B e13 B i13 B l13 B a14 B b14 B c14 B d14 B g14 B i14 B l14 B m14 B n14"
/>
Try it on [https://hexworld.org/board/#14c1,:pa14:pb14:pc14:pd14:pg14:pi14:pl14i12m14n12n14g11e13m11i13g10l13l10g12g9h12l9k12g8j11l8i10g7h9l7k9g6j8l6i7g5h6l5k6g4j5l4i4g3i3l3k3g2h2l2k1 HexWorld].

=== Puzzle 5 ===
Red to move and Blue to win.

<hexboard size="14x14"
coords="show"
contents="B n1 B m2 R n2 B l3 R m3 R n3 R k4 R l4 B m4 R n4 B e5 B f5 B g5 B h5 B i5 B j5 B k5 B l5 R m5 B c6 R e6 R f6 R g6 R h6 R i6 B j6 R k6 R l6 R c7 R d7 B e7 B f7 B g7 B h7 B i7 B k7 R l7 R b8 B c8 B d8 R e8 R f8 R g8 R h8 B i8 R b9 R d9 B e9 R f9 B g9 R h9 R c10 B d10 R e10 B f10 R g10 B h10 R c11 B d11 R e11 B f11 R g11 R h11 R i11 B b12 B c12 B d12 B e12 R f12 B g12 R j12 B k12 B g13 B h13 R j13 R k13 B e14 B i14"
/>
Try it on [https://hexworld.org/board/#14c1,:pe14:pi14j13g13k13h13f12b12j12c12c11d12e11e12g11g12h11k12i11d11c10f11e10d10g10f10b9h10d9e9f9g9h9c8b8d8e8i8f8e7g8f7h8g7c7h7d7i7l7k7e6c6f6j6g6e5h6f5i6g5k6h5l6i5m5j5k4k5l4l5n4m4m3l3n3m2n2n1 HexWorld].

== References ==
The puzzles are taken from:

* Claude Berge. L'Art Subtil du Hex. Manuscript, 1977.

via

* [[Jack van Rijswijck]]. Set Colouring Games, 2006.
* [[Ryan Hayward]]. Berge and the Art of Hex, 2003.
== See also ==

* Main article: [[Puzzles]]
* [[Solutions to Claude Berge's puzzles|Solutions]]

[[category:Puzzle]]

User:Hexanna

2025-12-13T16:39:48Z

Hexanna: /* Ladders */ add diagram

[[Openings on 19 x 19]]

[[Strategic advice from KataHex]]

I started learning Hex mid-July of 2021, about 2 weeks before I created my LittleGolem account.

[[PlayHex]] > Settings > Shading pattern > Custom > <code>t = (2*row + 3*col) % 7; (t==0 ? 0 : (3^t % 7))/6</code>

[https://github.com/hexanna1/hex-scripts Hex scripts]

==draft of more strategy stuff==

There's a fair amount of stuff for [[Strategic advice from KataHex]] that I've either been too lazy to write up / polish, or haven't thought of a good way to explain. I also haven't properly conveyed which things are important; some things come up rarely and should either be mentioned briefly or cut for the article. There's also a tradeoff in generality/specificity and concreteness/abstractness.

I need a lot more concrete examples, but it's hard to come by truly instructive examples (or know what people find useful)

It's a good idea to first study patterns in a '''vacuum''', meaning in the absence of nearby stones. This provides a useful baseline when evaluating a move. Local patterns can occur in "open" areas (areas far away from corners/edges, whether in a vacuum or not) as well as near corners/edges. Let's start with open patterns.
* ...

<hexboard size="3x3" float="inline" edges="none" coords="none" contents="R b1 c1 B -:b2 a3 E *:c2" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 c1 d2 B b3 S b1 b2 c2 d1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a3 c2 B d1 S b2 b3 E *:(c1 d2)" />
<hexboard size="3x4" float="inline" edges="none" coords="none" contents="R a2 b3 B b1 S b2 a3 E *:(a1 d2)" />

In these patterns, the shaded cells are expected to be empty.
* Pattern 1 is strong for Red (and weak for Blue), because Red threatens to play at (*) to kill the blue stone at (-). Even if Red hasn't played (*), the pattern is still weak for Blue, because the effectiveness of (-) is hindered by the other blue stone.
** This pattern frequently happens near the obtuse corner, as a result of bad minimaxing. In the first example below, Blue 3 is a common, but misguided, attempt at minimaxing. However, it's a mistake, and Red 4 (which creates the pattern) is the best reply. This position is weak for Blue ''even though Red does not plan to play (*)'' (as it is too close to his own edge) — Blue 3 hinders Blue 1, and the damage has already been done. In the second example, Blue 5 is a bad minimaxing move that creates the pattern, and Red 6 kills the blue stone.
*: <hexboard size="5x5" float="inline" edges="bottom left" coords="none" contents="B 1:d2 R 2:c3 B 3:e1 R 4:c2 E *:d3"
/> <hexboard size="6x6" float="inline" edges="bottom left" coords="none" contents="B 1:f1 R 2:d2 B 3:d3 R 4:e2 B 5:d4 R 6:c3" />

===Ladders===

* Ladders are tricky. Here's a first-order approximation — there are many exceptions, but this is still a useful baseline:
** As the defender of the ladder, you have two main options: push, or let your opponent connect in exchange for territory. Look ahead to see whether pushing seems good for you.
*** Generally, yield if and only if pushing would allow your opponent to adjacent-cut through two of your stones (which happens if you have a stone above the ladder).
*** If the attacker fails to push, you should either bridge or play adjacently on the attacker's row to cut in front of the ladder. In other words, Blue should play at (*) or (+) below:
***: <hexboard size="4x5" float="inline" edges="bottom" coords="none" contents="R arrow(12):a2 B a3 R b2 B b3 E *:d2 +:c2" />
** As the attacker of the ladder, your main options are to push or pivot. If you're close to an acute corner, you can either pivot, or if the corner is empty, play a cornering move.
*** Most of the time, pushing is good until you reach a corner.
*** If you have stones above the ladder, try to make use of them, mostly by jumping and pivoting early enough so that the stones aren't "wasted".

===Advanced strategy: Leaving the question open===

An important skill is learning when to tenuki. One class of examples is when you have a choice between two moves (or two sets of moves A and B), and playing one prevents you from playing the other one later. Your opponent is asking a [[question]]: do you want to play set A or set B? It can be advantageous to leave both possibilities open, so that if your opponent plays a move near the region, you can respond with the more advantageous reply of A and B. (The "dual" viewpoint: It can be a mistake to answer the question immediately — that gives your opponent a better idea of what to play in that region, both practically and theoretically. Answering it relieves your opponent of the obligation to play a move that counters ''both'' A and B.)

Here's an instructive example. Consider this 6-5 acute corner joseki ([https://hexworld.org/board/#19nc1,o14o15n15m17p16n17q14q15p15p17o17p14r12 HexWorld link]):
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d4 R 3:c4 B 4:b6 R 5:e5 B 6:c6 R 7:f3 B 8:f4 R 9:e4 B 10:e6 R 11:d6 B 12:e3 R 13:g1 S c5 d5 E A:(f2 c7) B:(g2 c8) *:a5 -:d7" />
* There is a lot to unpack. First, some general commentary before we talk about the "question".
** Blue 6 captures the two shaded hexes and should be an automatic reply to Red 5 for a strong player.
** Blue 10 is not the only possible reply after Red 9. (*) is another strong option. (-) looks locally tempting (it lets Red connect in exchange for a 2nd-row ladder escape) but it's a mistake, because it's redundant with Blue 8 which already provides a 2nd-row ladder escape.
** Red yields the soon-to-be ladder on move 13 to prevent an adjacent allow-cut with Red 1.
* After Red 13, it's now Blue to move. Blue has some reasonable-looking options in the hexes marked A or B. Which of these moves should Blue play, if any?
** Blue's plan is to either play both moves marked A or both moves marked B, but ''not'' one of each. Why?
** Ideally, if it worked, Blue would independently like to play A at the top (it's the more useful intrusion) and B at the bottom (it's better to defend a ladder than to let your opponent connect outright). But unfortunately for Blue, the top and bottom aren't independent. Blue can't do both, because if Blue plays A at the top (and Red defends the bridge at B, creating a capped flank), the ladder escapes at the bottom right corner are not strong enough for Blue to play B at the bottom; since Red will connect bottom-right regardless in this situation, Blue has to make the most of it by gaining some territory (at the bottom A) while letting Red connect, which Red was going to do regardless.
** In essence, Red has created a question for Blue: do you want to play the A set or the B set? There are cases where each option is better, but importantly, Blue has the deciding power here — Red cannot meaningfully force Blue to "settle" the question without otherwise making a concession. This "dual threat" means Red has to plan for the ''worst'' of A and B when making moves in the region. ''[A concrete example here would be nice.]''
** If Blue were to commit to one of the sets now, she would give up this dual threat for free. So, Blue should actually tenuki rather than playing either the A or B sets. (Indeed, KataHex thinks playing either set is an inaccuracy. In this particular example, Blue can also play at (*) instead of playing in a totally separate region.)
** Taking a step back, Red played move 13 (the locally optimal move) knowing that Blue would get the deciding power. The lesson is ''not'' that giving your opponent the deciding power is a huge disadvantage (such that you should make lots of concessions to avoid doing so); rather, it's important to be aware when a question exists and realize that settling it prematurely could be a mistake.

There are a couple other 6-5 acute corner joseki worth discussing, as they both come up frequently in real games. They are cases where move order matters, and the correct move order allows a player to neutralize the opponent's deciding power. Here is the first example:
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R 1:d3 B 2:d5 R 3:e5 B 4:d6 R 5:e6 B 6:e4 E *:c5 +:f2 A:b7 B:c7" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:f2 B 8:c7 R 9:f4 B 10:f3 R 11:g2 B 12:g3" />
<hexboard size="8x8" float="inline" edges="bottom right" coords="none" contents="R d3 B d5 R e5 B d6 R e6 B e4 R 7:c5 B 8:b7 R 9:f2 B 10:f4 E a:c7 b:g3" />
* In the first diagram, Red would like to play both (*) and (+). Both moves bridge efficiently from Red 1 while blocking Blue's stones.
** However, Blue has two possible ways to extend her group at the bottom, by playing either A or B. This is a question for Blue.
** Moving at (+) first would be a mistake because Blue has a strong minimaxing reply at B (see the second diagram). The best Red can do now is let Blue connect in the corner in exchange for territory at Red 11.
** Instead, Red should play at (*) first. Blue could reasonably respond at B, but A is slightly better (see the third diagram). However, this leaves a bridge intrusion at (a), and Red now has the ladder escape that he needs for Red 9 to threaten to connect (via (b)). Blue 10 still connects in the corner, but the position here is more favorable for Red than in the second diagram.

Here is the other joseki, where the question is more subtle (and less important):
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R 1:b2 B 2:b4 R 3:c4 B 4:b5 R 5:d5 B 6:c6 R 7:d6 E *:c3 +:c5" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c5 R 9:d4 B 10:c3 E A:d1 B:d2" />
<hexboard size="7x6" float="inline" edges="bottom right" coords="none" contents="R b2 B b4 R c4 B b5 R d5 B c6 R d6 B 8:c3 R 9:d1 B 10:c5" />
* In the first diagram, Blue would like to play both (*) and (+).
** If Blue plays (+) first (see the second diagram), after Blue 10, Red now has the deciding power to play either A or B. The tradeoff is that A lets Blue connect outright in exchange for slightly more useful territory (since it's a bridge away from Red 1), while B holds Blue to a 2-4 parallel ladder.
** Blue achieves a better outcome by playing (*) first (see the third diagram). Red must play A now, since if Red were to play B, then Blue would simply cut through and connect anyways. By playing in the correct move order, Blue practically strips Red of the deciding power between A and B. In some cases, this hardly matters and Red was going to play A regardless, but in general (*) is the safer move for Blue.

Finally, [https://playhex.org/games/bc0bf856-fe69-4088-a11b-1a700089866a here] is a game where Blue used this concept to make sense of the board and prune out some bad moves.
* The first relevant position is [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9,j9k8j8k7j5j6g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 17]. Blue would ideally like to convert the 4-6-8-10-12 group into a third-row ladder escape by playing g13, creating a capped flank, but Red can respond with the minimaxing m8 which deprives Blue 2 (the swap stone) of its 3rd-row ladder escape. Red asks Blue the question: "Do you want the third-row escape on the left edge and the second-row escape on the right edge, or vice versa?"
* As a further complication, [https://hexworld.org/board/#14nc1,g12:sd11c12d5d12e11e12f11f12g11g12i11h11i9i10k9j9k8j8k7j5j6,g3e6h4b6f2e3c2f3g2g4h3e2g6d8f8i6i4:rb after Red 23], Red asks Blue another question: "How do you eventually want to intrude in my 21-23 bridge? Do you want to play k6, which works well with Blue 22 and the swap stone to form a '6th row' ladder escape, or j7 to connect with your main group?" This question is not independent of the first one — if Blue gives up the third-row escape on the swap stone, then k6 no longer works well, and conversely, if Blue insists on keeping the third-row escape on the swap stone, then j7 makes Blue 22 a bit of a waste.
* With these two questions top of mind, Blue decided to tenuki on move 24 (and later). KataHex thinks this was a good decision, and settling any of the questions prematurely would've been a blunder. Furthermore, Blue attempted to tenuki in such a way that the decision power of the two questions was preserved (with debatable success), while hoping Red would make it easy for Blue to decide how to answer the questions. (Despite the game ending in a win for Blue, that hope never really materialized.)

==bullet hex strategy?==

I have not thought about this much, but you can model bullet Hex (i.e., Hex with very fast time controls) as Hex where some fraction of your moves must be premoves. Practically speaking, what does this mean? You should occasionally play premove tricks (moves that give you a large advantage assuming your opponent premoves the "expected" move or otherwise doesn't react to your move), and you should play in a way that's resilient to premove tricks, even if the moves are suboptimal in regular Hex. The frequency of premove tricks should increase significantly as players get into time pressure.

It would be interesting to study "bullet Hex strategy". Here's a starting example. Suppose you're Red and pushing the ladder below, but it's Blue's turn. Both players are under time pressure, and you have a split second to play a premove (or hover your mouse) if you wish. You ''expect'' Blue to continue pushing the ladder by playing at P. Where do you play?

<hexboard size="5x8"
coords="hide"
edges="bottom left"
contents="R b3 c3 d3 B b4 c4 E *:e3 +:f2 -:g3 P:d4 Q:e4 Y:d5 Z:f4"
/>

I think it depends:
* Case 1: The rest of the board is settled, Red has a forced win (for example, a ladder escape), and he simply needs to carry out that win without running out of time.
** Naively, Red just keeps pushing the ladder by premoving or hovering over (*). However, the risk here is that Blue can play a premove trick—she can yield by playing at Y instead of the expected P, and if Red premoves the push at (*) (or doesn't premove, but fails to react that Blue played an unexpected move and plays (*) nonetheless), then Blue plays at Z.
** A better idea for Red is to ''premove'' at P. Then, if Blue plays "as expected" by pushing the ladder at P (instead of yielding), the premove doesn't go through and Red plays at (*). However, if Blue does yield at Y, Red's premove goes through and is a good move.
* Case 2: There is play remaining elsewhere, so Red could conceivably let Blue connect in exchange for territory. Or, Red has a forced win but is behind on time, and the ladder is really long and he'll lose on time if he simply tries the strategy from Case 1.
** Most of the time, Red can still play the "premove P, click (*)" strategy from Case 1. However, with a small probability, Red waits for Blue to play at P, then he plays at (+) instead, hoping Blue will premove her ''next'' move. This is a premove ''trick'', and unlike the strategy from Case 1, it's important that Red doesn't play it 100% of the time—he needs to play it as part of a mixed strategy. If Blue notices, the downside is that the ladder has just been pushed up one row. But if Blue premoves and pushes the ladder again at Q, then Red plays at (-) and immediately wins.
** Blue might not like the fact that Red has this premove trick. One possible resource for her is to "premove (*), click Q" while defending the ladder; this probably works quite well in bullet.

==katahex/general strategy (draft material)==

Some miscellaneous topics — I've written rough drafts/outlines of a few but they aren't ready to be published. If you have KataHex, you might consider these "select topics" as an advanced study guide for large boards, to explore on your own.
* Common mistake: playing two adjacent stones (of the same color), either in a vacuum or in certain inefficient configurations.
** Closely related to "bad minimaxing". Chances are, if you want to play an adjacent stone and it's not (i) provably optimal, (ii) part of a joseki, or (iii) part of a forcing sequence, you're better off with a bridge move.
* Isosceles triangle pattern - this appears in lots of different josekis. The basic pattern is Red 5/7/9 in https://hexworld.org/board/#19nc1,c2d16b17e17c15b18e14e18d12, where the base of the "isosceles triangle" is perpendicular to your opponent's edge.
** As with all patterns, try to learn/intuit not only when it's efficient to play the pattern, but when it's inefficient to do so. (For instance, in a vacuum, it's inefficient to play two red stones perpendicular and close to the blue edge; you should play at a 30 degree angle instead. This pattern needs neighboring stones to be efficient.)
* Tom's move is useful for connecting, but it leaves several weaknesses and intrusion points if you don't absolutely have to connect.
** Useful starter position with a 2-4 parallel ladder for analysis: https://hexworld.org/board/#19nc1,e6e4d4e3c3d5c7d2f4f5e5g3f3g1f2f1e2e1d3c2b3b2g4h3g2h1
** Obtuse corner opening joseki with a 2-4 parallel ladder: https://hexworld.org/board/#19c1,a19j5d15c17c14c15b17b18a18d14a16d13
* Acute corner joseki continuations. Red 7 is an inaccuracy but this is a common joseki: https://hexworld.org/board/#19nc1,e6e4d4e3d3d5c7f2b7c5b10i2. Study the "continuation" moves Red 11 and Blue 12, and study similar moves in related joseki.
** For example, gain intuition for why Red 13 is the best move in https://hexworld.org/board/#19nc1,d5d4c5c3b4b3e4f2c4d3g3:pb8 (a5/b5 are captured by Red; see [[Theorems_about_templates#The_shape_of_templates]]), why Red 17 is the best move in https://hexworld.org/board/#19nc1,e6e5d6d4c5c4f5g3d5e4h4i2b5b4j3:pc9, and why other moves are weak.

===a3 opening===
Nowadays, strong players rarely play a3 on the first move because it's similar to c2 (which is already on the losing side on most board sizes), but a bit weaker. There are some subtle differences between a3 and c2, however, and most strategy advice on the a3 opening is outdated.

If Red plays a3, some older strategy guides recommend that Blue respond d6 or e7 if she wants to play in the acute corner. However, KataHex considers this an inaccuracy; even though d6 and e7 are attacking stones in the sense that Blue would win a [[bridge ladder]] towards the acute corner, they don't do a good job blocking Red's edge the way e4 does in an empty corner.

If you want to play the acute corner, 5-5, or 4-4 on smaller boards, is a good choice. (There are other options further away from the corner on larger boards.) KataHex and other bots consistently prefer these moves:
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2175191&nmove=4
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2033243&nmove=2
* https://www.littlegolem.net/jsp/game/game.jsp?gid=2018841&nmove=8
* https://www.littlegolem.net/jsp/game/game.jsp?gid=1950209&nmove=20

One reason may be the efficiency of templates [[Edge template IV2e|IV2e]] and [[Fifth_row_edge_templates#V-2-d|V2d]], both of which have a "hole"; the template can be rotated so that the hole coincides with Red's a3 stone. This is explored in more detail below.

It can be useful to realize that a3 is primarily a blocking move (blocking Blue on the left rather than helping Red connect to top), whereas c2 is half blocking and half connecting. To illustrate the point, it's a bad idea for Red to play both c2 and 4-4 obtuse corner on the top edge (in the absence of Blue's blocking stones), because that's too much strength near the top. However, Red could feasibly play both a3 and 4-4 on the top edge.

a3 can combine well with c5. Red's goal is to leave b4/c5 open as options (these are the same hexes used in the a3 escape trick, but of course, a3+c5 can also connect upwards). Blue should also keep an eye on b4 - not to play immediately because it's too close to her own edge - but as the most efficient option for connecting to the left side near the top. Blue would much prefer to connect using b4 (which is a locally efficient move that neutralizes a3's switchback and upwards connecting potential) than b5 (which only slightly weakens a3's potential, forcing Red to play d5 instead of c5), or a 2nd row ladder with b3 (which removes Red's weakness at b1).

a3 is one of my favorite openings to analyze (I only wish it were more fair as an opening move). Unlike c2 where Blue's best move is typically to tenuki, a3 gives Blue the option to fight in the same acute corner. There is essentially a separate theory of a3 corner joseki.

I find a3 lines particularly educational because a3 changes the corner just enough to make different moves more efficient. You can learn a lot about how an existing stone influences neighboring stones.

* d4: https://hexworld.org/board/#14c1,a3d4e4e3
** Note that Red e4 is near his own edge, but it's adjacent to Blue's stone at d4 and does a good job at blocking it; this is analogous to the joseki https://hexworld.org/board/#14nc1,k10k9, where Blue 2 is often a strong move. Blue e3 is counterintuitive, but it forms [[Fifth_row_edge_templates#V-2-d|template V2d]], which has a hole at Red's a3. In essence, Blue makes Red's a3 stone nearly useless in this line.
** Red avoids b5 which is a very weak move, because it makes Blue b4 even more efficient: https://hexworld.org/board/#14c1,a3d4b5b4c4d2. This position is practically Blue's dream; Blue makes full use of d4 and the weakness at a3 with [[Edge template IV2e|template IV2e]] (which also has a hole).
*** This is similar to Red 5 being a blunder in this opening: https://hexworld.org/board/#14nc1,a10d11c12c11b12b11
* e3 first: https://hexworld.org/board/#14c1,a3e3c4b4d2c3d3c5
* d3 is bad: https://hexworld.org/board/#14c1,a3d3b4
* Why is e4 weak? After Red c5!, the normally locally efficient b4 (even though it's globally inefficient, because it's on Blue's 2nd row) is terrible because Red d3 is a bridge cut. Note that a3 primarily is a blocking move, not a connecting one, so d3+a3 is not so much strength near the top that it renders d3 a bad move. https://hexworld.org/board/#14c1,a3e4c5b4d3
* e5: In some ways, these resemble the low intrusion 5-4 joseki, shifted up one row.
** https://hexworld.org/board/#14nc1,a3e5c5c6d6d5 Red 5 is optional but actually an acceptable bridge peep, because intruding on the other side of the bridge would be too much strength near the top and almost never a good idea.
** Red must still avoid b5 for the same reason as before: https://hexworld.org/board/#14nc1,a3e5c5c4b5b4d4f3d3 is very favorable to Blue. Observe that Blue 8 at f3 is more efficient than at e2, because e2 is followed by Red f3 which is a bridge cut
** https://hexworld.org/board/#14nc1,a3e5c5c4d4c6d6e6d5d7b8c7a7b6a6b5a5a4b4c2c3f3d2 Red peeps the bridge with 7. d6; Blue 8. e6 may be a stronger reply than connecting directly. While this variation represents strong play by both sides, and Red is doing just fine, Blue 20 and the sequence leading up to it is a good illustration of how Blue can play efficiently around Red's a3 stone.
** https://hexworld.org/board/#14nc1,a3e5c5d4c4c6f5h3 There are too many variations after Blue d4 to list them all. Blue 6 at d3 leads to a fun minimaxing sequence but is perhaps a slight inaccuracy: https://hexworld.org/board/#14nc1,a3e5c5d4c4d3b6d2c3d7

==Notation for distances==
When analyzing patterns, it's useful to have a way to express various distances concisely. There are terms like adjacent, bridge, "classic block," but larger distances are hard to describe. Distances like "two hexes away" are ambiguous, does it mean a distance of 2 (like a1 and c1) or two hexes in between (like a1 and d1)? Everyone has their own way of thinking about it. Here's the one I use:
* Pretend that the hex grid consists of regular hexagons, and adjacent hexagons' centers have distance 1. (Equivalently, the centers of the hexes form a triangular lattice; pretend these triangles are equilateral with side length 1.)
* It's not hard to show that a "bridge" (like a1 and b2) has length sqrt(3) if you connect the centers, and a "classic block" (like a1 and c2) has length sqrt(7). A fun exercise is to show that the distances between the centers of two hexes is always the square root of an integer. Which integers are possible as squared distances? Answer: [https://oeis.org/A003136 Löschian numbers.]
* The fact that squared distances are integers is notationally convenient. Let Δn denote a ''squared'' distance of n (I like this because it looks nicer than a square root symbol, and a letter like "d" can be confused with a coordinate).
* "Adjacent" is Δ1; "bridge" is Δ3; "classic block" is Δ7; the distance between a1 and c1 is Δ4; "two bridges away" is Δ12.
* Are there collisions? [https://oeis.org/A118886 Yes,] but the first collision is Δ49 (a distance of 7), which is both the distance between a1 and h1, and between a1 and f4. This is large enough that for practical purposes, the notation is unique. (An interesting fact: the first number with a 4-way collision is, coincidentally, [https://oeis.org/A198775 1729] of [https://en.wikipedia.org/wiki/1729_(number)#As_a_Ramanujan_number Ramanujan fame.])
* Here is a mapping of squared distances from the hex marked (*). Most of the time, you only need distances up to Δ16.
: <hexboard size="5x9"
coords="hide"
edges="hide"
visible="area(a1,e5,i1)"
contents="S a1
E *:a1 1:b1 3:b2 4:c1 7:c2 9:d1 12:c3 13:d2 16:e1 19:d3 21:e2 25:f1
27:d4 28:e3 31:f2 36:g1 37:e4 39:f3 43:g2 49:h1
48:e5 49:f4 52:g3 57:h2 64:i1"
/>

==Why I like the swap rule==

# The standard reason: It makes the game much more fair.
# Every Hex player knows "Hex without swap is a first-player win." Add the swap rule, and you get another elegant result for free: "Hex with swap is a second-player win."
# It provides much-needed asymmetry to the opening. The Hex board has two-fold rotational symmetry, and on odd-sized boards without swap, most players will open in the center. The symmetry isn't broken until the second turn!
# The opening stone is especially influential in determining the character of the game. On 19x19, games that open with c2 look quite different from those that open with a10. A lot of cool tactics, like the a3 and a4 switchbacks, only really appear when you add the swap rule. The swap rule greatly expands the variety of plausible positions, and who doesn't like more variety?
# While the second player has a theoretical advantage, the first player has a practical advantage: she can choose what opening to play. Because the opening stone is much more influential than later stones, the first player can pick an opening that she's especially familiar with, or one that her opponent is weak at. It would not be surprising if the first and second player's advantages cancel out among strong humans (on 19x19), and each player wins almost exactly 50% of games.
# Since the swap rule is nonconstructive, it scales up beautifully to larger boards. Even if players have no idea which opening stones are fair, it's virtually certain that fair stones exist, and there should be no qualms about fairness. Contrast this with komi in Go: a fair komi has to be empirically determined for each board size, and there could be disagreements about the komi for various board sizes. (This concern is alleviated with auction komi, but that adds another rule to the game.)
# It's resistant to partial solutions. Let's say that someone miraculously comes up with a constructive proof that the obtuse corner is winning on 19x19. No problem, players will just stop playing obtuse corner and play other opening moves instead.
# It's fun to try to trick players into swapping a weak move or not swapping a deceptively strong move. :)

==Insights and tidbits from KataHex (hzy's bot)==

* Swap map for 19×19 generated with net 20220618, with around 15k visits for most moves. For the red hexes, the number corresponds to Blue's Elo advantage if she swaps Red's move; for the blue hexes, the number corresponds to Blue's Elo advantage if she does not swap Red's move. Smaller numbers correspond to fairer openings. Includes all fair openings as well as a few selected unfair openings, including the strongest move without swap (e10). I used more visits for the fairest moves: 1000k for e3 (49.5% win rate), 500k for n3 (49.2%), 100k for a15 (52.5%).

: <hexboard size="19x19"
coords="show"
contents="S red:all blue:(a1--r1 a2--q2 a3 e3--n3 s3)
blue:(a17 f17--o17 s17 c18--s18 b19--s19)
E 65:(d3 p17)
3:(e3 o17)
73:(f3 n17)
76:(g3 m17)
73:(h3 l17)
84:(i3 k17)
90:(j3 j17)
103:(k3 i17)
104:(l3 h17)
49:(m3 g17)
6:(n3 f17)
47:(o3 e17)
59:(p3 d17)
121:(h4 l16)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
138:(m4 g16)
122:(i5 k15)
108:(j5 j15)
133:(k5 i15)
69:(q2 c18)
163:(p2 d18)
96:(b17 r3)
201:(b18 r2)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
146:(d2 p18)
100:(a3 s17)
137:(b3 r17)
157:(c3 q17)
83:(a4 s16)
73:(b4 r16)
136:(a5 s15)
93:(a6 s14)
95:(a7 s13)
131:(a8 s12)
99:(a9 s11)
41:(a10 s10)
81:(a11 s9)
115:(a12 s8)
78:(a13 s7)
56:(a14 s6)
17:(a15 s5)
57:(a16 s4)
110:(a17 s3)
174:(a18 s2)
56:(a19 s1)
382:(e10 o10)"
/>

==Random unsolved questions==

Most of these are very difficult to answer, and I would be happy if even a few were answered in the next few years:

* Is the obtuse corner always winning on larger board sizes? What about the b4 opening? Let P(n) be the statement that "the obtuse corner is a winning opening in n×n Hex without swap." There are a few possible cases; an interesting exercise is to come up with subjective probabilities of each case being true.
** A. P(n) is always true. If so, can we prove this?
** B. P(n) is true for infinitely many n, with finitely many counterexamples. If so, what's the smallest counterexample?
** C. P(n) is true for infinitely many n, with infinitely many counterexamples. If so, does P(n) hold "almost always," "almost never," or somewhere in between?
** D. P(n) is true for finitely many n. If so, what's the largest such n?
* Kriegspiel Hex (Dark Hex), a variant with incomplete information
** Under optimal mixed strategies, what is Red's win probability on 4×4?
** For larger boards (say, 19×19), is Red's win probability close to 50%?
*** If so, a swap rule might not be needed for Kriegspiel Hex, which would be neat.
*** If not, imagine a variant where Red's first move is publicly announced to both players, and Blue has the option to swap it. Which initial moves are the fairest now?

replies by [[User:Demer|Demer]]:

* https://zhuanlan.zhihu.com/p/476464087 has percentages, although it doesn't translate these into a guessed swap map and I don't know anything about the bot they came from.
** It suggests that [on 13x13, g3 is the most balanced opening] and [on 14x14, g3 should not be swapped].
** On 27x27 without swap, it likes the 4-4 obtuse corner opening slightly more than anything else nearby.
* As far as I'm aware, even 3×4 Dark Hex has not been solved. (https://content.iospress.com/articles/icga-journal/icg180057 apparently gives "some preliminary results" for that size.)

hexanna:

* Thank you, this is amazing! From the Google Translate, the bot is an adaptation of KataGo trained on 13×13 and smaller, using transfer learning to train larger nets on top of the 13×13 net for a short period of time. I may edit the [[swap rule]] article later with some insights.
** The results for up to 15×15 look very reliable to me. This is because many of the subtle patterns suggested by other bots, like leela_bot, appear in these swap maps. For example, on [https://pic3.zhimg.com/v2-53e66f72eb7129d5ffe676ae293ad856_r.jpg 13×13]:
*** a1–c1 are stronger than d1; a2–c2 ≥ d2 ≥ e2 in strength; and a similar relation holds for moves on the third row. See [[Openings on 11 x 11#d2]].
*** b4 is weaker than all of its neighbors, because Blue can fit the ziggurat in the corner.
*** j3 is surprisingly weak and i3 is surprisingly strong. Many people were surprised about this when leela_bot's swap map came out, but the result may be more than just random noise.
*** a10 is the weakest of a4–a10, while a5 is the strongest.
*** b10 is stronger than all of its neighbors, because Blue cannot fit the ziggurat in the obtuse corner.
** That this bot picked up on all these subtleties, and assigns a win percentage close to 100% for most moves on 13×13, suggest to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, and the author seems to agree, the 37×37 map looks very unreliable. I see percentages as low as 37% but only as high as 54% (for a move like f1, which should almost certainly be a losing move).
** The 27×27 map looks more reliable. I'm personally very skeptical that moves on Red's 6th row are among the most balanced moves, but there are some interesting (if somewhat noisy) insights to be had still.

==Recursive swap==
Not really a serious suggestion, just for fun. One advantage of "recursive swap" over multi-stone swap is that opening preparation plays a smaller role, because both players have control over the first n stones.

RECURSIVE_SWAP'[k, depth, color]:
if depth = 0:
[color] continues playing as normal.
else:
[color] plays a move. [~color] can either
swap[k], or
RECURSIVE_SWAP'[k+1, depth-1, ~color]

RECURSIVE_SWAP[n]:
RECURSIVE_SWAP'[1, n, Red]

RECURSIVE_SWAP[0]:

Red continues playing as normal.

RECURSIVE_SWAP[1]:

Red plays a move. Blue can either
* swap, or
* continue playing as normal.

RECURSIVE_SWAP[2]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** continue playing as normal.

RECURSIVE_SWAP[3]:

Red plays a move. Blue can either
* swap, or
* play a move, after which Red can either
** swap2, or
** play a move, after which Blue can either
*** swap3, or
*** continue playing as normal.

===Analysis===
RECURSIVE_SWAP[0] is the same as playing with no swap.

RECURSIVE_SWAP[1] is the same as playing with the swap rule.

RECURSIVE_SWAP[2]:
* Red shouldn't play a move that's too strong or it'll be swapped.
* If Red plays a weak stone, Blue should try to play a move just strong enough that Red will be indifferent to swap2. (If a "fair" move is half a stone, and Red plays a weak move worth x < 0.5 stones, Blue should play a move worth x + 0.5 stones.)
* Red should try to play a weak move that's also hard for Blue to equalize (so that Red gets a sizable advantage when deciding whether to swap2 or not).

RECURSIVE_SWAP[3]:
* If Red plays a move worth x > 0.5 stones, Blue should swap.
* If Red plays a weak stone worth x < 0.5, Blue should play a move worth less than x + 0.5 (or else Red will swap2). If Blue's move is worth y, then Red should play a move as close to (y - x) + 0.5 as possible, so that Blue's swap3 decision is difficult.
* Red should play a weak move that's hard for Blue to find a tricky reply to (where a "tricky" reply is one that makes it hard for Red to equalize, such that Blue has an easy time deciding whether to swap3 or not).

User:Hexanna

2025-12-13T14:56:02Z

Hexanna: hexworld -> hexboard converter

Strategic advice from KataHex

2025-12-13T14:25:24Z

Hexanna: more formatting/compress spacing

==Introduction==

Several hex bots have popped up in the last few years. Many are neural nets and adaptations of Go bots like AlphaGo Zero. The most recent (and likely strongest) as of 2025 is [[KataHex]], which is based on the open-source KataGo. These bots have influenced human play in many aspects already; one of the most notable is the bots' preference for the 4-4 obtuse corner over the 5-5. Many old strategy guides have become partially (if not mostly) outdated.

This article attempts to bring some strategy advice "up-to-date". There's a lot of random insights; most are a direct result of playing around with KataHex evaluations for thousands of hours, including analyzing the games of strong players for common mistakes. Since this is strategic advice and not tactics, the advice won't apply 100% of the time, and many players may disagree with some parts. Win percentages and evaluations in this article are based on KataHex net [https://github.com/hzyhhzy/KataGo/releases/tag/Hex_20240812 20240812], which is significantly stronger than previous nets.

Care has been taken to only draw conclusions when there are multiple examples of KataHex preferring something. However, the main goal is to give actionable advice to make players stronger, and there isn't rigorous proof for basically any of these statements.

===Big picture, and some meta-points===

* '''TL;DR:''' What are the most important principles? Even if you learn nothing else, you should learn these:
*# Avoid playing close to your own edge, especially in the opening, unless you have a very good reason to do so.
*#* Think of your own edge as a whole row of stones of your color. You already have a bunch of stones (and thus strength) there, and playing yet another one close by would be a waste. That stone is much better served elsewhere, where you don't have as much strength.
*# Bridges are the most efficient way to connect your own stones—not adjacent stones, trapezoids, crescents, or hammocks. If there are no opposing stones nearby, playing adjacent to an existing stone of your color is usually a mistake; if you want to do that, consider whether there's a better move a bridge away from that stone.
*#* If there are nearby opposing stones, things become more nuanced, but bridging to your existing stones is often still a good idea.
*# Usually, cut through an opponent's bridge if given the opportunity, and avoid letting it happen to you. In the pattern below, Red cuts through Blue's two stones a bridge apart. This is very good for Red and bad for Blue, because it's very hard for Blue to connect the two stones now.
*#: <hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 c1 R b1 1:b2" />
* This guide assumes you've read [http://www.mseymour.ca/hex_book/hexstrat.html ''Hex: A Strategy Guide''] or have equivalent knowledge.
** ''Hex: A Strategy Guide'' predates KataHex, but it's very well made, and 90% of the advice is still good.
* On smaller boards, tactics and raw calculation matter more. On larger boards, strategy matters more. (It's ultimately a spectrum, and it varies by skill level: for beginners, size 13 might be considered large; for strong players, tactics can matter a great deal on size 13.)
** On smaller boards, the exact board size affects strategy (some moves can be good on 11×11 but not 12×12). The larger the board, the less the exact board size matters. The general strategy for 19×19 and 20×20 should be nearly identical.
** On smaller boards, there is more space close to corners and edges. On larger boards, there is more "open" space (hexes not close to corners and edges). I intentionally said "open" space and not "center" space, because there is not much special about hexes near the center of the board, except that they are far away from corners and edges. But on large boards, there are many such areas.
** In particular, the larger the board size (starting at around size 13–15), the more important it is to know how to play in "open" areas. Many existing guides cover corner/edge strategy heavily—and correctly so, as corners/edges remain important up to very large sizes (25–30 or so). People read those guides and often know how to play the first 10–20 moves of a 17×17 game, but often not after that, when most of the game is still remaining!
** Strategy can also be ''local'' or ''global''. Local strategy tells you how to play within a small region of the board, and global strategy tells you how to play in relation to the whole board (or across multiple local regions). Both are important.
** The strategy of "open" areas is mostly local, and so the larger the board, the higher the relative importance of local strategy. Why?
*** On a large Hex board, the "middle" is relatively homogeneous—two separate open areas near the middle are basically the same in terms of how you should play them; it doesn't matter that one of the areas is 15 hexes away from an edge and the other one 18. Because of this homogeneity, the same basic principles will apply multiple times in the span of a game.
*** Knowing how to play across multiple local regions is very hard, and it's often different across games (especially for humans). It's better to master local play first, since similar local play principles will be relevant every game.
* How should a game of Hex (on larger boards) broadly feel like?
** While the win condition is binary (you either connect or you don't), the actual gameplay feels like a gradual accumulation of tiny advantages and disadvantages. Strong players often "know" they're winning in the middlegame, not because they see an actual way to connect, but because their opponent made some mistakes, however small, along the way. (You rarely know for sure who's winning until the end, and you're ultimately making an educated guess.)
** In particular, it's usually very hard to tell who's winning just by looking at a position. Contrast that with games like chess, where you can often tell by counting up material and broadly looking at positional factors. In Hex, it's much more tractable to tell how good your position is ''relative to a few moves ago.''
** Relatedly, there's (almost) no meaningful notion of "long-term strategy" in Hex. If anything, it can be harmful to create a long-term plan and stubbornly stick to it, because the moves you should play often differ greatly depending on your opponent's responses.
** Hex strategy is equally what to play and what ''not'' to play. The latter is frequently underemphasized. Each stone is worth a lot, and even one bad move can "undo" 10–20 good ones. The risk-reward balance strongly favors avoiding mistakes, rather than attempting risky but potentially "brilliant" moves.
* Now some points about how to learn. From what I observe, people often read strategy advice and then fail to follow it consistently. This is often either because of skepticism, or not realizing just how often particular pieces of advice should be followed (it's tempting to think that you've found the exception to the rule).
** It's healthy to be skeptical—there is a lot of ''bad'' strategy advice, and it would probably do more harm than good to implement every piece of advice you come across. So I'm not expecting readers to take everything in this guide at face value either (and many statements have nuances/exceptions, some of which I don't even realize should be explained to readers; it's hard to translate precise intuitions into words after all). Unfortunately, separating the signal from the noise is hard. If you're skeptical about a principle or how strongly it holds, try running it through KataHex yourself (if you have it installed), and try variations to understand the nuances, when the principle holds, and importantly ''why'' it holds. Sometimes people are skeptical of KataHex itself or think they can do better; I think there are good heuristic reasons why this thinking is misguided (the most simple one is that KataHex is far stronger than the strongest humans).

==General==

===Common mistake: playing near your own edge===

Playing near your own edge with no nearby stones is a bad idea in the opening and even middlegame. This is one of the most common mistakes among human players. You need a strong (often tactical) reason to do otherwise.
* When is playing near your own edge a good idea? There are a few cases, like when you're playing adjacent to existing stones, responding to a joseki, or pushing a ladder as the attacker, but all of these involve stones in the vicinity.
* This holds most strongly for 13×13 to 19×19, but applies more often than not to 11×11 too.
* Playing near your own edge in the opening or early middlegame should be a "last resort". I would recommend never doing so in the first 30-40 moves in 19×19 unless you have a very good reason.

===Common mistake: bad minimaxing===

Many intermediate and strong players are too eager to minimax. They play bad minimaxing moves (that KataHex thinks are inferior to directly connecting) more often than they fail to minimax. The issue is that minimaxing isn't always free; your opponent can often intrude for useful territory, and sometimes the territory your opponent gains is more useful than what you gain by minimaxing. It's a difficult skill to judge which player gains more from a minimax, but there are common patterns.
* Players frequently attempt to minimax by playing adjacent to their existing stone on the 5th or 6th row of their own edge. Playing two adjacent stones like this is not very efficient and typically a mistake.
* Blue's minimaxing attempt is unsuccessful; KataHex already thinks Red is 98% to win: https://hexworld.org/board/#15nc1,c2e11c13e10b12e9c10d8b9c7a8b6c6b7d7c8e8d9
* A common example; Red is 98% to win: https://hexworld.org/board/#13nc1,c2d10c11c10b11c9b8a10. Note that Red 7 at b8 is stronger than at a10, because b8 combines well with c2 and b5 on 13×13. On 15×15, KataHex prefers 7. a12 (the equivalent of a10 on 13×13) instead.
* A case where minimaxing is in fact the best move: https://hexworld.org/board/#13nc1,c2d10c9d11

===Common mistake: bad bridge peep===

A common scenario is the [[Peep#Bridge_peep | bridge peep]]. Blue should play at (*) as long as she thinks it's forcing enough for Red to defend his bridge.

However, even if Blue hasn't played (*) yet, Red should usually mentally place a blue stone there. Especially on larger boards, Blue will be able to play at (*) sometime in the future. This is important: even if (*) is empty, I like to think of this pattern as ''implying'' a blue stone at (*) (in the absence of tactics that somehow prevent Blue from playing there). Another way (useful for me, but the analogy might be nonsense to others) to think about it is that Blue's stone at A is well-placed enough relative to Red's bridge that she has some "potential energy" at (*).

<hexboard size="3x3" coords="hide" edges="hide" contents="R c1 b3 B A:a2 E *:c2" />

However, 80% of the time, if there's no blue stone at A, Blue should not intrude on either side of the bridge. Intruding is usually a mistake because it settles the [[question]] for Red. If Red knows he will have a stone at A or B because Blue already intruded in the other spot, Red can better plan his future moves because of that! There are exceptions; if Blue is confident that only one of the intrusions will ever be useful for her, then intruding is often strong.

<hexboard size="3x2" coords="hide" edges="hide" contents="R b1 a3 E A:a2 B:b2"/>

===Ladders===
* Don't worry about switchbacks, climbing, or ladder creation templates on 15x15 or larger. They can be theoretically interesting and make for good puzzles, but games are rarely close enough for these to matter. I personally don't think about them on any board size, but others may reasonably disagree with this approach.
* Q: When should the defender of a ladder yield? A: Usually, don't yield unless your opponent can cut through two of your stones, one above and one below the ladder. In that case, consider yielding.
** [https://hexworld.org/board/#13nc1,c2e9c10d8b9c11d10d12 Here] is an example. If Blue instead plays 8. d11, Red 9. e10 cuts through e9 and d11.
** In [https://playhex.org/games/66b8a125-bf02-42d2-8900-af3042a8ccf8 this game] from the [[Tournaments#Hex_Monthly | Hex Monthly 21]] ([https://hexworld.org/board/#14c1,c2d11k10k4b5f10g9i9i8g10h9h10j8j9b12c10l8k8l7k7l6k11j11h12m5k6l4m4l5l9k9h13i13l3k5i5j5j4i6h6f8h7i7f9g8f7e8e7d8d7b8c8b9c9a11b13d10c11c12,c13d12d13e11:rw HexWorld link]), the yielding move 60. c14 is winning, while the played 60. c13 (which allows Red d12, cutting through Blue's c13 and d11 stones) is losing.

===Other===

* Be aggressive; always be ready to capitalize on an opponent's mistake. Here are a few illustrative examples of this and some of the above concepts:
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2368410&nmove=8 this game], 6. c9 was a mistake (and an example of incorrect minimaxing); 7. a10 is the [[Openings_on_19_x_19#4-4_obtuse_corner | standard reply]]. White should have played 8. b8; j4 was a mistake. Black's 9. b8 is the "capitalizing" move; KataHex already says 99% win rate for Black. In this case, there were several good options on move 9. These alternatives are threatening enough that Black didn't have to play b8, but this won't always be true, like in the next example.
** [https://littlegolem.net/jsp/game/game.jsp?gid=2362913&nmove=26 Here] is another example (interestingly, White resigned in a winning position according to the bot, though the final position is highly tactical). KataHex actually thinks Black has only an 35% win rate after 26. m6. However, 27. d11 is a "minimaxing with adjacent stones" mistake, bringing Black's win rate down to 3%.
** [https://www.littlegolem.net/jsp/game/game.jsp?gid=2400947&nmove=6 Here] is an example where White's win rate increased consistently from 65% after move 6 to 98% after move 16. The aggressive bridging sequence in moves 8–16 is very strong because it basically neutralizes Black's stones on d16 and d17; it is a big mistake for Black to ignore the threat here. Move 11 is also an example of a bad bridge peep.

* Don't play a move that makes your opponent's existing stones unnecessarily well-placed relative to your new stone. This is covered briefly in a couple places in the [[Openings_on_19_x_19#General_principles | 19×19 opening guide]]. Here are a couple more examples.
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2369954&nmove=26 this game] ([https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6 HexWorld link]), what should Black play on move 27? White is threatening to connect 26 back to the 16–24 group if Black isn't careful. Black's obvious options are to play at c8 and c9. Which option is better?
** A tempting answer is c8 because it appears to connect more closely to the top. However, note that White's stone on move 14 functions exactly like an opening a3 stone. For example, it can escape 2nd but not 3rd row ladders. If Black plays c8, White can play the [[A3 escape trick]], as shown in [https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6c8e7d8e8d9e9 this continuation]. This sequence allows White to "make the most" of stone 14, and Black shouldn't allow it. Better for Black is 27. c9, which doesn't allow White to carry out a 3rd-to-5th row switchback. Indeed, KataHex evaluates c9 as 97% to win for Black and c8 as only 30%.
** Consider [https://hexworld.org/board/#13n,c2d10c11c10b11e6b10d8l4l3c8d5b5d6h7h6j5i6k7 this position] (which is based on [https://www.littlegolem.net/jsp/game/game.jsp?gid=2369782 this game]). Where should White play in the lower-right acute corner? Black had just played at the "3-7" point from White's perspective; as mentioned in the 19×19 opening guide, a strong response to 3-7 is 4-4 (at j10) instead of the usual 5-4 (at j9). A player who is short on time can play j10 without thinking too hard. As it turns out, KataHex thinks j10 is the best move, evaluating it as 91% for White, whereas j9 is only around 50%.

* Sometimes, you know you have a completely winning position after your opponent blunders very early in the game. Your goal is to preserve that win until the end of the game. Don't overextend yourself and try to win too "quickly", possibly making possibly suboptimal moves in the process. Even when KataHex plays itself with a significant handicap (like playing without swap), a significant fraction of the board seems necessary to carry out the win, assuming strong defense. I prefer taking my time to "fill the board" by focusing on playing moves at least as good as my opponent's, instead of worrying about connecting in as few moves as possible.

==Local patterns==

An especially effective way to improve at Hex is to learn and gain intuition about lots of small local patterns. These patterns are small enough that they occur basically everywhere when you play Hex. Knowing the patterns can help you avoid blunders (for example, avoiding adjacent or bridge ''allow-cuts'', as explained below) and make good moves when it's otherwise unclear what to play.

===Cuts===
<hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 c1 R b1 1:b2" />
<hexboard size="1x3" float="inline" edges="none" coords="none" contents="B a1 c1 R 1:b1" />
<hexboard size="3x4" float="inline" edges="none" coords="none" visible="-(a1 b1 d1 a3 c3 d3)" contents="B a2 d2 R c1 1:b3 S b2 c2 E *:(b2 c2)" />
<hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a3 c1)" contents="B a1 c3 R 1:b2 S b1 a2 c2 b3" />
Consider the four patterns above. Let's call the patterns '''adjacent cut''', '''near cut''', '''short bridge cut''' (or just '''bridge cut'''), and '''long bridge cut''' respectively. In some sense, Red is "cutting" through two of Blue's stones in each pattern, so Red is the '''attacker''' and Blue is the '''defender'''. We can also use '''cut''' as an attacking verb and '''allow-cut''' as a defending verb. For example, the adjacent cut pattern is favorable for the attacker (Red above), so we can say that it's a good idea to ''adjacent-cut'' through opponent stones, and a bad idea to ''adjacent allow-cut'' (meaning it's a bad idea to play a move that allows your opponent to adjacent-cut through two of your stones).

Why is it strong to adjacent-cut (and equivalently, why is it a mistake to adjacent allow-cut)? The most intuitive reason is that if Blue allow-cuts, it takes 3 extra moves for her to connect her two stones, as shown below. In a sense, this makes it unlikely for both stones to be concurrently useful in the final connection — if one of the stones proves useful, chances are the other one won't be.
<hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 c3)" contents="B a2 c1 1:a3 2:b3 3:c2 R b1 b2" />

The (short) bridge cut is strong for the attacker (Red above) because he captures the two hexes marked (*). If Blue allow-cuts, often the damage has already been done even if Red doesn't actually cut through immediately on the next turn. (However, note that this isn't always true; some tactical situations require Red to cut through immediately instead of playing elsewhere.)
* There are situations where Blue ''should'' allow-cut. If Red's cutting move comes with concessions, like the move is on his second or third row (and especially if Red already has a stone near his own edge, so cutting through would be overplaying that edge), the pattern is much less favorable to Red.

The near cut and long bridge cut are also favorable for the attacker, but much less clearly so — there are many more "exceptions" where cutting is bad and allow-cutting is fine. On the other hand, the adjacent and short bridge cuts are quite strong, and cutting through is usually the best move (unless doing so involves playing a stone that's too close to your own edge, or another major concession).

Note that allow-cuts can occur not only ''proactively'' before the attacker plays the cutting move, but also ''retroactively'' after the "cutting" move is played (of course, it's not really a "cutting" move until after the defender allow-cuts). In other words, Blue 1 is considered an allow-cut in both of these examples (proactive on the left and retroactive on the right), and they are equally bad for Blue assuming that Red intends to cut in the proactive case:
<hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 1:c1 R b1" />
<hexboard size="2x3" float="inline" edges="none" coords="none" visible="-(a1 c2)" contents="B a2 1:c1 R b1 b2" />

An important way of pruning moves and avoiding blunders is making sure you don't play moves that adjacent or bridge allow-cut, either proactively or retroactively, unless you have a good reason to do so. In particular, the retroactive case can be easy to overlook if you're not used to thinking in this way.

===Small patterns===
<hexboard size="2x2" float="inline" edges="none" coords="none" visible="-a1" contents="B b2 R b1 1:a2" />
<hexboard size="3x3" float="inline" edges="none" coords="none" visible="-(a1 a3 c2 c3)" contents="B b3 R c1 1:a2" />
<hexboard size="4x2" float="inline" edges="none" coords="none" visible="-(a1 b4)" contents="B a4 R 1:b1" />

In each of these patterns, we will discuss the strength of Red's move 1 ''in a vacuum'', meaning in the absence of nearby stones, like near the middle of an empty board on 19×19 or even larger boards. It is most useful to analyze each pattern without the influence of nearby stones or edges (which can be thought of as virtual rows of stones).
* Pattern 1: It's typically an inaccuracy for Blue to play adjacent to a red stone in a vacuum. Red's best response is usually playing adjacent to both stones, as in Red 1.
* Pattern 2: Blue plays a bridge away from Red's stone. This is often a good move, and Red 1 a bridge away from both stones is also good.
* Pattern 3: Red 1 is a classic block. This is often fine for Red, and Blue doesn't have an obvious best local response. Blue could tenuki here.

===Capture patterns===

Know the most common [[captured cell]] patterns. It really reduces the number of moves you have to consider.

The bridge cut is one of the most important capture patterns.

* A typical example can be found in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2233736&nmove=10 this game]. After 10. g11, Black would have the easiest time with 11. h9, with 95% win rate, instead of the played k8 with 74% win rate. After k8 was played, one of White's best moves would've been h9 to prevent Black from playing the same. KataHex does think Black has other reasonable options on move 11, but they mostly appear to just delay h9.

Here is another common capture pattern. The other patterns on [[captured cell]] are useless because they have too many Red stones adjacent to each other, which is too inefficient to occur in normal play.

<hexboard size="3x3" coords="hide" edges="hide" contents="R c1 c3 b3 B a2 S b2 c2 E *:(b2 c2)" />

This ''one-sided'' capture pattern is very common. I like to mentally replace A with a blue stone and (*) with a red stone (even though Blue won't always achieve this), because Blue should never play at (*). [[Edge template IV3a]] is useful to know here.
<hexboard size="3x2" coords="hide" edges="hide" visible="-a1" contents="R b1 a3 b3 S a2 b2 E *:b2 A:a2" />

The trapezoid and crescent also one-sided-capture the hexes at (*). Anecdotally, this fact is more useful on smaller boards like 11×11, but it's less common overall than the previous capture patterns.
<hexboard size="3x3" float="inline" coords="none" edges="none" visible="-a1" contents="R b1 a3 c3 c1 S a2--c2 b3 E *:(c2 b3)" />
<hexboard size="3x3" float="inline" coords="none" edges="none" visible="-a1" contents="R a2 b1 c1 c3 S b2 c2 a3 b3 E *:c2" />

==How to think about the opening and middlegame==

Some of these points sound like hyperbole, but they are a fairly accurate representation of how I think about Hex, especially on larger boards. There are exceptions, but exceptions are rarer than one might think.

* In the first half of the game, focus all of your attention into playing "efficient" moves. These are moves that work well with existing stones and board edges. Excellent intuition of efficient patterns is enough to get strong middlegame positions against even top players (as of 2024).
* Don't worry about tactics until the late middlegame. It's a waste of time to analyze sequences >10 moves deep. In the opening, the best move you can find with only intuition is no worse than the best move you can find with calculation.
** Why? Compared to bots, humans are very bad at calculation. Your intuition (which you learn directly from KataHex) is more useful than hypothetical hours or days of error-prone human analysis.
* Use your knowledge of efficient patterns to plan ahead. This is where you want to do shallow analysis, typically 1-5 moves deep. A good move does as much as possible:
** allows you to play efficient moves later
** prevents your opponent from playing an efficient move outright
** allows your opponent an efficient move, but only with concessions (say, a locally efficient but globally inefficient move, or a move that's efficient with relative to one stone but inefficient relative to another)

These points require a lot of elaboration. What do I mean by "efficient"? (Not to be confused with [[efficiency]].) There are two broad notions, '''global''' and '''local''' efficiency.
* Globally efficient stone: Broadly, any move that isn't too close to your own edge. Moves that are strong in the absence of nearby stones (like on an empty board). Corner moves are very efficient. On 19×19, moves near your opponent's 5th row are very efficient.
* Locally efficient stone: In the presence of nearby stones, some configurations are broadly "good" or "bad" for one player. They aren't ''always'' good or bad because of tactics, but because humans can't calculate everything, it's extremely useful to have decent heuristics that work most of the time.

In a sense, you want to play the move that maximizes the "sum" of local and global efficiency. This isn't an exact science; use your intuition to make an educated guess.
* The best move locally is rarely the best move globally. Hex is a game of concessions and tradeoffs. There is no "free lunch," unless you count concepts like inferiority/domination.
* Conversely, if it looks like your opponent is beating you on one side of the board, there are two possibilities:
*# You played suboptimal moves in that region.
*# Your opponent overplayed that region, which necessarily means they made concessions elsewhere (usually the opposite side of the board). This means your opponent ''underplayed'' the other region.
* If option 1 is true, you are probably losing anyways, so assume (or hope) option 2 is true. If you opponent overplays region A and underplays region B, it's typically a good idea to focus your play in region B. Why? If you keep playing in region A, you're potentially helping your opponent out of their mistake, by making your opponent less over-connected in region A. For example, your opponent could have a minimaxing move that makes them much stronger in region B, at the "concession" of making them weaker in region A — but this isn't really a concession because your opponent had strength to spare in region A.

The discussion so far has been highly abstract. What does efficient play look like in practice? As explained earlier, it's strong to adjacent-cut and bridge-cut through your opponent's stones. The small patterns in the "Capture patterns" section above are also locally efficient for the capturing player. Of course, just as offense and defense are equivalent in Hex, so are playing locally efficient moves and inducing your opponent to play locally ''inefficient'' moves.

In the following examples, P1 and P2 are efficient patterns for Red relative the corresponding edges, while P3 is inefficient. P1 is common when Red opens with a first column opening. P2 requires elaboration: normally Red 1 is a weak move because it's too close to Red's own edge. However, when Red must play on his first 3 rows (because of very limited space), Red 1 or its mirror image at (*) is usually the best, because it prevents Blue from playing there and forming the equivalent of P1 herself. P2 also occurs in theoretical contexts, like [[Template Va#If Blue moves at y:]]. In P3, Red 1 is a tempting (because of [[edge template III1b]]), common, but inefficient move, because Blue's response at one of (*) is usually strong. If Red must play near his own edge, he should instead play as in P2 above.
P1: <hexboard size="3x2" float="inline" edges="left" coords="none" contents="R a2 1:b3" />
P2: <hexboard size="2x6" float="inline" edges="bottom" coords="none" contents="B c2 R 1:e1 E *:b1" />
P3: <hexboard size="3x5" float="inline" edges="bottom" coords="none" contents="B c3 R 1:d1 E *:(b2 e2)" />

===Making inferences: An extended example===

Use the patterns and intuitions that you know, to reason and make inferences in unfamiliar positions. You might not always find the best move, but you can often prune bad moves. Here is an extended example, involving several closely related patterns that are useful to know in isolation.
# ''Pattern:'' If you play the 4-4 3-3 obtuse corner joseki, it is important to know that Blue 6 [https://hexworld.org/board/#19nc1,d16c17d17c18d18e14c15 here] is weak.
#* This is because Red 7 undermines Blue 6 and also reduces Blue 2/4 from a 3rd row ladder escape to a 2nd row escape. Also, Blue's potential intrusion at c16 doesn't gain her anything.
#* Instead, [https://hexworld.org/board/#19nc1,d16c17d17c18d18d15 here] is a good local response for Blue. This gives Blue a 5th row ladder escape while blocking Red 1.
#* On the other hand, if Blue only has a 2nd row escape and doesn't have the "potential energy" to get a 3rd row escape, you might guess that Blue 6 above becomes a good local move, because Red 7 loses one of its main strengths. Such a guess would be correct: in [https://hexworld.org/board/#19nc1,d16c17p15d17e16g16e17f18d18e14c15 this position], Blue 10 is a strong move. On the other hand, Blue 10 at d15 would be weak here, as it's quite wasteful, only converting a 2nd row escape into a 3rd row escape.
# ''Pattern:'' If Red tenukis on move 3, then Blue 4 is locally strong in [https://hexworld.org/board/#19nc1,d16e14:pf15 this position]. In particular, Blue 4 is much stronger than bridging directly towards Blue's own edge with g13, because the former is a strong blocking move that reduces the effectiveness of Red 1. This strength is in spite of Red's intrusion on move 5. (I first saw this move in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2090091&nmove=16 this bot game] from [https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=739 this thread on LittleGolem].)
#* Variations of this pattern appear frequently in obtuse corner joseki. If Red started with the 5-5 obtuse corner, then the equivalent Blue 4 is [https://hexworld.org/board/#19nc1,e15f13:pg14 also] strong.
#* Another simple example is Red 5 in [https://hexworld.org/board/#19nc1,d16d15f14:pe13 this position].
# ''Inference:'' Let us consider [https://hexworld.org/board/#19nc1,d16c17d17c18d18:p this position] again. Suppose Blue tenukis on move 6. What is a good local response for Red on move 7? There is more than one right answer, but you just need to find one move that isn't a blunder.
#* ''Hint:'' Use the two patterns above to come up with an answer.
#* ''Hint:'' Consider that Blue has a 3rd row ladder escape, so Blue d15 is locally strong but Blue e14 is weak. This is the case for future blue stones in this local region, even though Blue already played elsewhere on move 6. Therefore, Red should perhaps try to play a move that makes Blue d15 less attractive, even if it comes at the concession of making Blue e14 more attractive (since the latter is ''a priori'' a weak move, such a concession gives up less than it otherwise might).
#* ''Answer:'' Red 7 [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13 here] is a good local move.
#** Why? Ignoring stones 2-5, the presence of Red 1 and 7 makes Blue e14 more attractive. Red's 4-4 stone, much like the obtuse corner opening, tends to shift Blue's efficient stone locations up one row. With that in mind, Red's move 7 is a bit like Red e14 on an empty board, where Blue e15 is [https://hexworld.org/board/#19nc1,e14e15 the best response]. However, stones 2-5 make Blue e14 a bad move, despite Red 7 making it slightly less bad (this is the concession-but-not-really-a-concession that Red intended).
#** Let's now consider Blue d15. Normally a decent local move after stones 1-5, it becomes quite weak after Red 7, because of [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13d15f14 this sequence]. This is a typical example of the second pattern in this extended example. (Actually, if Blue had two moves in a row in this region, Blue 8 wouldn't be so bad because she could follow up with f14, where Red wants to play move 9. This is a tactical situation where it's bad for Red to tenuki.)
#** If we wanted to be complete, Blue actually has another option for move 8, f14 first which threatens d15: [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13f14d15d14e14 here]. This is an example of the "skew cut" from above. Red should respond with d15 himself, or else Blue can play d15. This is probably Blue's best local response.
#** Every move makes a concession, but some concessions are more effective than others. As alluded to in the hint, Red 7 weakens a strong Blue reply (d15) while strengthening a weak Blue reply (e14). Red's goal is to minimize the strength of Blue's ''best'' response (rather than the strength of Blue's average response), and Red 7 at e13 accomplishes that beautifully.

Part of getting stronger at Hex (as with other games) is being able to compress knowledge and patterns effectively. If you understand this extended example, you can infer the best move in many similar but novel situations, without memorizing each one individually. If you forget which move is exactly the best, you can recover the right move with high probability by just reasoning about related patterns that you do know are strong or weak. The "A3 escape trick" example earlier in this article is another instance of this concept.

On larger boards and particularly near the opening or middlegame, when you have a lot of time to think about a position, it may be worth spending a good chunk of it making inferences and scoping out moves that are either good or to be avoided by each side, rather than spending most of your time calculating tons of lines and variations.
* In fact, once you've scoped out that "moves A and B are good for Red, while moves X and Y are bad for Red" and similarly for Blue, that makes the process of analyzing and pruning lines much more efficient.

==Joseki==

* The 4-3 corner move is almost always a mistake compared to the 5-4, even on 11×11. If your opponent plays 4-3, 90% of the time the best local reply is 3-3. If 4-3 is played, here are some common sequences:
: <hexboard size="5x7" float="inline" coords="hide" edges="bottom right" contents="R 1:e2 B 2:e3 R 3:d3"
/> <hexboard size="5x7" float="inline" coords="hide" edges="bottom right" contents="R 1:e2 B 2:e3 R 3:f2 B 4:f3 R 5:d3 E a:b4 b:c5"
/>
: In the second joseki, Blue should play either (a) or (b), or she can defer the question until later. On larger boards with no nearby stones, (b) is often better.
* If your opponent plays the 5-4 joseki, a reasonable choice is to always play the 4-4 "high intrusion." KataHex prefers the 4-4 response by far, especially on 13×13 and larger. It's easily the "safest" choice, and many strong players have recently shown a clear preference for 4-4.
* In this often-played joseki, Red 7 is a mistake because of the territory Blue gets after move 8:
: <hexboard size="6x10" float="inline" edges="bottom right" coords="none" contents="R 1:g2 B 2:g3 R 3:h2 B 4:h4 R 5:i3 B 6:i4 R 7:f3 B 8:e5 R 9:d4 B 10:c6 R 11:b5 B 12:c4 R 13:c5" />
:* If the obtuse corner on Red's same edge is empty, a more principled response for Red is to play in that corner, like this:
:: <hexboard size="6x14" float="inline" edges="bottom left right" coords="none" contents="R 1:k2 B 2:k3 R 3:l2 B 4:l4 R 5:d3" />
:* If the obtuse corner is already occupied by Blue, then Red should still respond in the obtuse corner. One reasonable option (among others) is this:
:: <hexboard size="6x14" float="inline" edges="bottom left right" coords="none" contents="B d3 R 1:k2 B 2:k3 R 3:l2 B 4:l4 R 5:e3" />
* See more discussion of acute corner joseki at [[Openings_on_19_x_19#Acute_corner_theory]]. For the obtuse corner, see [[Openings_on_19_x_19#Obtuse_corner_theory]] which covers responses to the 4-4 and 5-5 obtuse corner in great detail. Most of this is applicable to 13×13 and larger, and not just 19×19.

==Specific tactics==
===Threatening a bridge cut===
One of my favorite tactics is threatening a bridge cut, where the opponent needs to play too close to their own edge to defend:
<hexboard size="8x8" float="inline" edges="bottom left" coords="hide" contents="B *:d5 g2 R 1:f4 B 2:e3" />
Red 1 is a strong move, because it forces Blue to respond at 2 (or else Red will play the bridge cut there), but Blue 2 is normally overplaying on the left side when combined with Blue's stone at (*). Be on the lookout for this when your opponent plays the 4-4 and 7-7 obtuse corner, or 5-5 and 8-8.

This tactic can work well in other configurations, like [https://hexworld.org/board/#19c1,:pg8:pg11h9 this] and [https://hexworld.org/board/#19c1,:pe9:ph9g8 this]. However, be careful when using this tactic! When Blue isn't actually making a concession by bridging her two stones, like when the pattern occurs further from her own edge, it can be a blunder for Red to threaten a bridge cut. [https://hexworld.org/board/#19nc1,:pg8:pj5i7h6 Here] is an example where Red 5 is a blunder in the absence of nearby stones.

[https://www.littlegolem.net/jsp/game/game.jsp?gid=2436000&nmove=12 Here] is an example from a real game. Black's best move is 13. h5 (in the actual game, Black doesn't play this until move 19 but is still arguably rewarded for doing so). One might think that h5 is too strong near the top edge because of c2 and b5, but this is not the case because the white stone on c5 partially neutralizes b5.

[[category:strategy]]

User:Hexanna

2025-12-13T14:19:17Z

Hexanna: fix formatting

Strategic advice from KataHex

2025-12-13T14:16:55Z

Hexanna: /* Joseki */ replace hexworld with diagrams, "fix" formatting

==Introduction==

Several hex bots have popped up in the last few years. Many are neural nets and adaptations of Go bots like AlphaGo Zero. The most recent (and likely strongest) as of 2025 is [[KataHex]], which is based on the open-source KataGo. These bots have influenced human play in many aspects already; one of the most notable is the bots' preference for the 4-4 obtuse corner over the 5-5. Many old strategy guides have become partially (if not mostly) outdated.

This article attempts to bring some strategy advice "up-to-date". There's a lot of random insights; most are a direct result of playing around with KataHex evaluations for thousands of hours, including analyzing the games of strong players for common mistakes. Since this is strategic advice and not tactics, the advice won't apply 100% of the time, and many players may disagree with some parts. Win percentages and evaluations in this article are based on KataHex net [https://github.com/hzyhhzy/KataGo/releases/tag/Hex_20240812 20240812], which is significantly stronger than previous nets.

Care has been taken to only draw conclusions when there are multiple examples of KataHex preferring something. However, the main goal is to give actionable advice to make players stronger, and there isn't rigorous proof for basically any of these statements.

===Big picture, and some meta-points===

* '''TL;DR:''' What are the most important principles? Even if you learn nothing else, you should learn these:
*# Avoid playing close to your own edge, especially in the opening, unless you have a very good reason to do so.
*#* Think of your own edge as a whole row of stones of your color. You already have a bunch of stones (and thus strength) there, and playing yet another one close by would be a waste. That stone is much better served elsewhere, where you don't have as much strength.
*# Bridges are the most efficient way to connect your own stones—not adjacent stones, trapezoids, crescents, or hammocks. If there are no opposing stones nearby, playing adjacent to an existing stone of your color is usually a mistake; if you want to do that, consider whether there's a better move a bridge away from that stone.
*#* If there are nearby opposing stones, things become more nuanced, but bridging to your existing stones is often still a good idea.
*# Usually, cut through an opponent's bridge if given the opportunity, and avoid letting it happen to you. In the pattern below, Red cuts through Blue's two stones a bridge apart. This is very good for Red and bad for Blue, because it's very hard for Blue to connect the two stones now.
*#* <hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 c1 R b1 1:b2"
/>
* This guide assumes you've read [http://www.mseymour.ca/hex_book/hexstrat.html ''Hex: A Strategy Guide''] or have equivalent knowledge.
** ''Hex: A Strategy Guide'' predates KataHex, but it's very well made, and 90% of the advice is still good.
* On smaller boards, tactics and raw calculation matter more. On larger boards, strategy matters more. (It's ultimately a spectrum, and it varies by skill level: for beginners, size 13 might be considered large; for strong players, tactics can matter a great deal on size 13.)
** On smaller boards, the exact board size affects strategy (some moves can be good on 11×11 but not 12×12). The larger the board, the less the exact board size matters. The general strategy for 19×19 and 20×20 should be nearly identical.
** On smaller boards, there is more space close to corners and edges. On larger boards, there is more "open" space (hexes not close to corners and edges). I intentionally said "open" space and not "center" space, because there is not much special about hexes near the center of the board, except that they are far away from corners and edges. But on large boards, there are many such areas.
** In particular, the larger the board size (starting at around size 13–15), the more important it is to know how to play in "open" areas. Many existing guides cover corner/edge strategy heavily—and correctly so, as corners/edges remain important up to very large sizes (25–30 or so). People read those guides and often know how to play the first 10–20 moves of a 17×17 game, but often not after that, when most of the game is still remaining!
** Strategy can also be ''local'' or ''global''. Local strategy tells you how to play within a small region of the board, and global strategy tells you how to play in relation to the whole board (or across multiple local regions). Both are important.
** The strategy of "open" areas is mostly local, and so the larger the board, the higher the relative importance of local strategy. Why?
*** On a large Hex board, the "middle" is relatively homogeneous—two separate open areas near the middle are basically the same in terms of how you should play them; it doesn't matter that one of the areas is 15 hexes away from an edge and the other one 18. Because of this homogeneity, the same basic principles will apply multiple times in the span of a game.
*** Knowing how to play across multiple local regions is very hard, and it's often different across games (especially for humans). It's better to master local play first, since similar local play principles will be relevant every game.
* How should a game of Hex (on larger boards) broadly feel like?
** While the win condition is binary (you either connect or you don't), the actual gameplay feels like a gradual accumulation of tiny advantages and disadvantages. Strong players often "know" they're winning in the middlegame, not because they see an actual way to connect, but because their opponent made some mistakes, however small, along the way. (You rarely know for sure who's winning until the end, and you're ultimately making an educated guess.)
** In particular, it's usually very hard to tell who's winning just by looking at a position. Contrast that with games like chess, where you can often tell by counting up material and broadly looking at positional factors. In Hex, it's much more tractable to tell how good your position is ''relative to a few moves ago.''
** Relatedly, there's (almost) no meaningful notion of "long-term strategy" in Hex. If anything, it can be harmful to create a long-term plan and stubbornly stick to it, because the moves you should play often differ greatly depending on your opponent's responses.
** Hex strategy is equally what to play and what ''not'' to play. The latter is frequently underemphasized. Each stone is worth a lot, and even one bad move can "undo" 10–20 good ones. The risk-reward balance strongly favors avoiding mistakes, rather than attempting risky but potentially "brilliant" moves.
* Now some points about how to learn. From what I observe, people often read strategy advice and then fail to follow it consistently. This is often either because of skepticism, or not realizing just how often particular pieces of advice should be followed (it's tempting to think that you've found the exception to the rule).
** It's healthy to be skeptical—there is a lot of ''bad'' strategy advice, and it would probably do more harm than good to implement every piece of advice you come across. So I'm not expecting readers to take everything in this guide at face value either (and many statements have nuances/exceptions, some of which I don't even realize should be explained to readers; it's hard to translate precise intuitions into words after all). Unfortunately, separating the signal from the noise is hard. If you're skeptical about a principle or how strongly it holds, try running it through KataHex yourself (if you have it installed), and try variations to understand the nuances, when the principle holds, and importantly ''why'' it holds. Sometimes people are skeptical of KataHex itself or think they can do better; I think there are good heuristic reasons why this thinking is misguided (the most simple one is that KataHex is far stronger than the strongest humans).

==General==

===Common mistake: playing near your own edge===

Playing near your own edge with no nearby stones is a bad idea in the opening and even middlegame. This is one of the most common mistakes among human players. You need a strong (often tactical) reason to do otherwise.
* When is playing near your own edge a good idea? There are a few cases, like when you're playing adjacent to existing stones, responding to a joseki, or pushing a ladder as the attacker, but all of these involve stones in the vicinity.
* This holds most strongly for 13×13 to 19×19, but applies more often than not to 11×11 too.
* Playing near your own edge in the opening or early middlegame should be a "last resort". I would recommend never doing so in the first 30-40 moves in 19×19 unless you have a very good reason.

===Common mistake: bad minimaxing===

Many intermediate and strong players are too eager to minimax. They play bad minimaxing moves (that KataHex thinks are inferior to directly connecting) more often than they fail to minimax. The issue is that minimaxing isn't always free; your opponent can often intrude for useful territory, and sometimes the territory your opponent gains is more useful than what you gain by minimaxing. It's a difficult skill to judge which player gains more from a minimax, but there are common patterns.
* Players frequently attempt to minimax by playing adjacent to their existing stone on the 5th or 6th row of their own edge. Playing two adjacent stones like this is not very efficient and typically a mistake.
* Blue's minimaxing attempt is unsuccessful; KataHex already thinks Red is 98% to win: https://hexworld.org/board/#15nc1,c2e11c13e10b12e9c10d8b9c7a8b6c6b7d7c8e8d9
* A common example; Red is 98% to win: https://hexworld.org/board/#13nc1,c2d10c11c10b11c9b8a10. Note that Red 7 at b8 is stronger than at a10, because b8 combines well with c2 and b5 on 13×13. On 15×15, KataHex prefers 7. a12 (the equivalent of a10 on 13×13) instead.
* A case where minimaxing is in fact the best move: https://hexworld.org/board/#13nc1,c2d10c9d11

===Common mistake: bad bridge peep===

A common scenario is the [[Peep#Bridge_peep | bridge peep]]. Blue should play at (*) as long as she thinks it's forcing enough for Red to defend his bridge.

However, even if Blue hasn't played (*) yet, Red should usually mentally place a blue stone there. Especially on larger boards, Blue will be able to play at (*) sometime in the future. This is important: even if (*) is empty, I like to think of this pattern as ''implying'' a blue stone at (*) (in the absence of tactics that somehow prevent Blue from playing there). Another way (useful for me, but the analogy might be nonsense to others) to think about it is that Blue's stone at A is well-placed enough relative to Red's bridge that she has some "potential energy" at (*).

<hexboard size="3x3"
coords="hide"
edges="hide"
contents="R c1 b3 B A:a2 E *:c2"
/>

However, 80% of the time, if there's no blue stone at A, Blue should not intrude on either side of the bridge. Intruding is usually a mistake because it settles the [[question]] for Red. If Red knows he will have a stone at A or B because Blue already intruded in the other spot, Red can better plan his future moves because of that! There are exceptions; if Blue is confident that only one of the intrusions will ever be useful for her, then intruding is often strong.

<hexboard size="3x2"
coords="hide"
edges="hide"
contents="R b1 a3 E A:a2 B:b2"
/>

===Ladders===
* Don't worry about switchbacks, climbing, or ladder creation templates on 15x15 or larger. They can be theoretically interesting and make for good puzzles, but games are rarely close enough for these to matter. I personally don't think about them on any board size, but others may reasonably disagree with this approach.
* Q: When should the defender of a ladder yield? A: Usually, don't yield unless your opponent can cut through two of your stones, one above and one below the ladder. In that case, consider yielding.
** [https://hexworld.org/board/#13nc1,c2e9c10d8b9c11d10d12 Here] is an example. If Blue instead plays 8. d11, Red 9. e10 cuts through e9 and d11.
** In [https://playhex.org/games/66b8a125-bf02-42d2-8900-af3042a8ccf8 this game] from the [[Tournaments#Hex_Monthly | Hex Monthly 21]] ([https://hexworld.org/board/#14c1,c2d11k10k4b5f10g9i9i8g10h9h10j8j9b12c10l8k8l7k7l6k11j11h12m5k6l4m4l5l9k9h13i13l3k5i5j5j4i6h6f8h7i7f9g8f7e8e7d8d7b8c8b9c9a11b13d10c11c12,c13d12d13e11:rw HexWorld link]), the yielding move 60. c14 is winning, while the played 60. c13 (which allows Red d12, cutting through Blue's c13 and d11 stones) is losing.

===Other===

* Be aggressive; always be ready to capitalize on an opponent's mistake. Here are a few illustrative examples of this and some of the above concepts:
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2368410&nmove=8 this game], 6. c9 was a mistake (and an example of incorrect minimaxing); 7. a10 is the [[Openings_on_19_x_19#4-4_obtuse_corner | standard reply]]. White should have played 8. b8; j4 was a mistake. Black's 9. b8 is the "capitalizing" move; KataHex already says 99% win rate for Black. In this case, there were several good options on move 9. These alternatives are threatening enough that Black didn't have to play b8, but this won't always be true, like in the next example.
** [https://littlegolem.net/jsp/game/game.jsp?gid=2362913&nmove=26 Here] is another example (interestingly, White resigned in a winning position according to the bot, though the final position is highly tactical). KataHex actually thinks Black has only an 35% win rate after 26. m6. However, 27. d11 is a "minimaxing with adjacent stones" mistake, bringing Black's win rate down to 3%.
** [https://www.littlegolem.net/jsp/game/game.jsp?gid=2400947&nmove=6 Here] is an example where White's win rate increased consistently from 65% after move 6 to 98% after move 16. The aggressive bridging sequence in moves 8–16 is very strong because it basically neutralizes Black's stones on d16 and d17; it is a big mistake for Black to ignore the threat here. Move 11 is also an example of a bad bridge peep.

* Don't play a move that makes your opponent's existing stones unnecessarily well-placed relative to your new stone. This is covered briefly in a couple places in the [[Openings_on_19_x_19#General_principles | 19×19 opening guide]]. Here are a couple more examples.
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2369954&nmove=26 this game] ([https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6 HexWorld link]), what should Black play on move 27? White is threatening to connect 26 back to the 16–24 group if Black isn't careful. Black's obvious options are to play at c8 and c9. Which option is better?
** A tempting answer is c8 because it appears to connect more closely to the top. However, note that White's stone on move 14 functions exactly like an opening a3 stone. For example, it can escape 2nd but not 3rd row ladders. If Black plays c8, White can play the [[A3 escape trick]], as shown in [https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6c8e7d8e8d9e9 this continuation]. This sequence allows White to "make the most" of stone 14, and Black shouldn't allow it. Better for Black is 27. c9, which doesn't allow White to carry out a 3rd-to-5th row switchback. Indeed, KataHex evaluates c9 as 97% to win for Black and c8 as only 30%.
** Consider [https://hexworld.org/board/#13n,c2d10c11c10b11e6b10d8l4l3c8d5b5d6h7h6j5i6k7 this position] (which is based on [https://www.littlegolem.net/jsp/game/game.jsp?gid=2369782 this game]). Where should White play in the lower-right acute corner? Black had just played at the "3-7" point from White's perspective; as mentioned in the 19×19 opening guide, a strong response to 3-7 is 4-4 (at j10) instead of the usual 5-4 (at j9). A player who is short on time can play j10 without thinking too hard. As it turns out, KataHex thinks j10 is the best move, evaluating it as 91% for White, whereas j9 is only around 50%.

* Sometimes, you know you have a completely winning position after your opponent blunders very early in the game. Your goal is to preserve that win until the end of the game. Don't overextend yourself and try to win too "quickly", possibly making possibly suboptimal moves in the process. Even when KataHex plays itself with a significant handicap (like playing without swap), a significant fraction of the board seems necessary to carry out the win, assuming strong defense. I prefer taking my time to "fill the board" by focusing on playing moves at least as good as my opponent's, instead of worrying about connecting in as few moves as possible.

==Local patterns==

An especially effective way to improve at Hex is to learn and gain intuition about lots of small local patterns. These patterns are small enough that they occur basically everywhere when you play Hex. Knowing the patterns can help you avoid blunders (for example, avoiding adjacent or bridge ''allow-cuts'', as explained below) and make good moves when it's otherwise unclear what to play.

===Cuts===
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 c1 R b1 1:b2"
/>
<hexboard size="1x3"
float="inline"
edges="none"
coords="none"
contents="B a1 c1 R 1:b1"
/>
<hexboard size="3x4"
float="inline"
edges="none"
coords="none"
visible="-(a1 b1 d1 a3 c3 d3)"
contents="B a2 d2 R c1 1:b3 S b2 c2 E *:(b2 c2)"
/>
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a3 c1)"
contents="B a1 c3 R 1:b2 S b1 a2 c2 b3"
/>
Consider the four patterns above. Let's call the patterns '''adjacent cut''', '''near cut''', '''short bridge cut''' (or just '''bridge cut'''), and '''long bridge cut''' respectively. In some sense, Red is "cutting" through two of Blue's stones in each pattern, so Red is the '''attacker''' and Blue is the '''defender'''. We can also use '''cut''' as an attacking verb and '''allow-cut''' as a defending verb. For example, the adjacent cut pattern is favorable for the attacker (Red above), so we can say that it's a good idea to ''adjacent-cut'' through opponent stones, and a bad idea to ''adjacent allow-cut'' (meaning it's a bad idea to play a move that allows your opponent to adjacent-cut through two of your stones).

Why is it strong to adjacent-cut (and equivalently, why is it a mistake to adjacent allow-cut)? The most intuitive reason is that if Blue allow-cuts, it takes 3 extra moves for her to connect her two stones, as shown below. In a sense, this makes it unlikely for both stones to be concurrently useful in the final connection — if one of the stones proves useful, chances are the other one won't be.
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c3)"
contents="B a2 c1 1:a3 2:b3 3:c2 R b1 b2"
/>

The (short) bridge cut is strong for the attacker (Red above) because he captures the two hexes marked (*). If Blue allow-cuts, often the damage has already been done even if Red doesn't actually cut through immediately on the next turn. (However, note that this isn't always true; some tactical situations require Red to cut through immediately instead of playing elsewhere.)
* There are situations where Blue ''should'' allow-cut. If Red's cutting move comes with concessions, like the move is on his second or third row (and especially if Red already has a stone near his own edge, so cutting through would be overplaying that edge), the pattern is much less favorable to Red.

The near cut and long bridge cut are also favorable for the attacker, but much less clearly so — there are many more "exceptions" where cutting is bad and allow-cutting is fine. On the other hand, the adjacent and short bridge cuts are quite strong, and cutting through is usually the best move (unless doing so involves playing a stone that's too close to your own edge, or another major concession).

Note that allow-cuts can occur not only ''proactively'' before the attacker plays the cutting move, but also ''retroactively'' after the "cutting" move is played (of course, it's not really a "cutting" move until after the defender allow-cuts). In other words, Blue 1 is considered an allow-cut in both of these examples (proactive on the left and retroactive on the right), and they are equally bad for Blue assuming that Red intends to cut in the proactive case:
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 1:c1 R b1"
/>
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 1:c1 R b1 b2"
/>

An important way of pruning moves and avoiding blunders is making sure you don't play moves that adjacent or bridge allow-cut, either proactively or retroactively, unless you have a good reason to do so. In particular, the retroactive case can be easy to overlook if you're not used to thinking in this way.

===Small patterns===
<hexboard size="2x2"
float="inline"
edges="none"
coords="none"
visible="-a1"
contents="B b2 R b1 1:a2"
/>
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 a3 c2 c3)"
contents="B b3 R c1 1:a2"
/>
<hexboard size="4x2"
float="inline"
edges="none"
coords="none"
visible="-(a1 b4)"
contents="B a4 R 1:b1"
/>

In each of these patterns, we will discuss the strength of Red's move 1 ''in a vacuum'', meaning in the absence of nearby stones, like near the middle of an empty board on 19×19 or even larger boards. It is most useful to analyze each pattern without the influence of nearby stones or edges (which can be thought of as virtual rows of stones).
* Pattern 1: It's typically an inaccuracy for Blue to play adjacent to a red stone in a vacuum. Red's best response is usually playing adjacent to both stones, as in Red 1.
* Pattern 2: Blue plays a bridge away from Red's stone. This is often a good move, and Red 1 a bridge away from both stones is also good.
* Pattern 3: Red 1 is a classic block. This is often fine for Red, and Blue doesn't have an obvious best local response. Blue could tenuki here.

===Capture patterns===

Know the most common [[captured cell]] patterns. It really reduces the number of moves you have to consider.

The bridge cut is one of the most important capture patterns.

* A typical example can be found in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2233736&nmove=10 this game]. After 10. g11, Black would have the easiest time with 11. h9, with 95% win rate, instead of the played k8 with 74% win rate. After k8 was played, one of White's best moves would've been h9 to prevent Black from playing the same. KataHex does think Black has other reasonable options on move 11, but they mostly appear to just delay h9.

Here is another common capture pattern. The other patterns on [[captured cell]] are useless because they have too many Red stones adjacent to each other, which is too inefficient to occur in normal play.

<hexboard size="3x3"
coords="hide"
edges="hide"
contents="R c1 c3 b3 B a2 S b2 c2 E *:(b2 c2)"
/>

This ''one-sided'' capture pattern is very common. I like to mentally replace A with a blue stone and (*) with a red stone (even though Blue won't always achieve this), because Blue should never play at (*). [[Edge template IV3a]] is useful to know here.
<hexboard size="3x2"
coords="hide"
edges="hide"
visible="-a1"
contents="R b1 a3 b3 S a2 b2 E *:b2 A:a2"
/>

The trapezoid and crescent also one-sided-capture the hexes at (*). Anecdotally, this fact is more useful on smaller boards like 11×11, but it's less common overall than the previous capture patterns.
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
visible="-a1"
contents="R b1 a3 c3 c1 S a2--c2 b3 E *:(c2 b3)"
/>
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
visible="-a1"
contents="R a2 b1 c1 c3 S b2 c2 a3 b3 E *:c2"
/>

==How to think about the opening and middlegame==

Some of these points sound like hyperbole, but they are a fairly accurate representation of how I think about Hex, especially on larger boards. There are exceptions, but exceptions are rarer than one might think.

* In the first half of the game, focus all of your attention into playing "efficient" moves. These are moves that work well with existing stones and board edges. Excellent intuition of efficient patterns is enough to get strong middlegame positions against even top players (as of 2024).
* Don't worry about tactics until the late middlegame. It's a waste of time to analyze sequences >10 moves deep. In the opening, the best move you can find with only intuition is no worse than the best move you can find with calculation.
** Why? Compared to bots, humans are very bad at calculation. Your intuition (which you learn directly from KataHex) is more useful than hypothetical hours or days of error-prone human analysis.
* Use your knowledge of efficient patterns to plan ahead. This is where you want to do shallow analysis, typically 1-5 moves deep. A good move does as much as possible:
** allows you to play efficient moves later
** prevents your opponent from playing an efficient move outright
** allows your opponent an efficient move, but only with concessions (say, a locally efficient but globally inefficient move, or a move that's efficient with relative to one stone but inefficient relative to another)

These points require a lot of elaboration. What do I mean by "efficient"? (Not to be confused with [[efficiency]].) There are two broad notions, '''global''' and '''local''' efficiency.
* Globally efficient stone: Broadly, any move that isn't too close to your own edge. Moves that are strong in the absence of nearby stones (like on an empty board). Corner moves are very efficient. On 19×19, moves near your opponent's 5th row are very efficient.
* Locally efficient stone: In the presence of nearby stones, some configurations are broadly "good" or "bad" for one player. They aren't ''always'' good or bad because of tactics, but because humans can't calculate everything, it's extremely useful to have decent heuristics that work most of the time.

In a sense, you want to play the move that maximizes the "sum" of local and global efficiency. This isn't an exact science; use your intuition to make an educated guess.
* The best move locally is rarely the best move globally. Hex is a game of concessions and tradeoffs. There is no "free lunch," unless you count concepts like inferiority/domination.
* Conversely, if it looks like your opponent is beating you on one side of the board, there are two possibilities:
*# You played suboptimal moves in that region.
*# Your opponent overplayed that region, which necessarily means they made concessions elsewhere (usually the opposite side of the board). This means your opponent ''underplayed'' the other region.
* If option 1 is true, you are probably losing anyways, so assume (or hope) option 2 is true. If you opponent overplays region A and underplays region B, it's typically a good idea to focus your play in region B. Why? If you keep playing in region A, you're potentially helping your opponent out of their mistake, by making your opponent less over-connected in region A. For example, your opponent could have a minimaxing move that makes them much stronger in region B, at the "concession" of making them weaker in region A — but this isn't really a concession because your opponent had strength to spare in region A.

The discussion so far has been highly abstract. What does efficient play look like in practice? As explained earlier, it's strong to adjacent-cut and bridge-cut through your opponent's stones. The small patterns in the "Capture patterns" section above are also locally efficient for the capturing player. Of course, just as offense and defense are equivalent in Hex, so are playing locally efficient moves and inducing your opponent to play locally ''inefficient'' moves.

In the following examples, P1 and P2 are efficient patterns for Red relative the corresponding edges, while P3 is inefficient. P1 is common when Red opens with a first column opening. P2 requires elaboration: normally Red 1 is a weak move because it's too close to Red's own edge. However, when Red must play on his first 3 rows (because of very limited space), Red 1 or its mirror image at (*) is usually the best, because it prevents Blue from playing there and forming the equivalent of P1 herself. P2 also occurs in theoretical contexts, like [[Template Va#If Blue moves at y:]]. In P3, Red 1 is a tempting (because of [[edge template III1b]]), common, but inefficient move, because Blue's response at one of (*) is usually strong. If Red must play near his own edge, he should instead play as in P2 above.
P1: <hexboard size="3x2"
float="inline"
edges="left"
coords="none"
contents="R a2 1:b3"
/>
P2: <hexboard size="2x6"
float="inline"
edges="bottom"
coords="none"
contents="B c2 R 1:e1 E *:b1"
/>
P3: <hexboard size="3x5"
float="inline"
edges="bottom"
coords="none"
contents="B c3 R 1:d1 E *:(b2 e2)"
/>

===Making inferences: An extended example===

Use the patterns and intuitions that you know, to reason and make inferences in unfamiliar positions. You might not always find the best move, but you can often prune bad moves. Here is an extended example, involving several closely related patterns that are useful to know in isolation.
# ''Pattern:'' If you play the 4-4 3-3 obtuse corner joseki, it is important to know that Blue 6 [https://hexworld.org/board/#19nc1,d16c17d17c18d18e14c15 here] is weak.
#* This is because Red 7 undermines Blue 6 and also reduces Blue 2/4 from a 3rd row ladder escape to a 2nd row escape. Also, Blue's potential intrusion at c16 doesn't gain her anything.
#* Instead, [https://hexworld.org/board/#19nc1,d16c17d17c18d18d15 here] is a good local response for Blue. This gives Blue a 5th row ladder escape while blocking Red 1.
#* On the other hand, if Blue only has a 2nd row escape and doesn't have the "potential energy" to get a 3rd row escape, you might guess that Blue 6 above becomes a good local move, because Red 7 loses one of its main strengths. Such a guess would be correct: in [https://hexworld.org/board/#19nc1,d16c17p15d17e16g16e17f18d18e14c15 this position], Blue 10 is a strong move. On the other hand, Blue 10 at d15 would be weak here, as it's quite wasteful, only converting a 2nd row escape into a 3rd row escape.
# ''Pattern:'' If Red tenukis on move 3, then Blue 4 is locally strong in [https://hexworld.org/board/#19nc1,d16e14:pf15 this position]. In particular, Blue 4 is much stronger than bridging directly towards Blue's own edge with g13, because the former is a strong blocking move that reduces the effectiveness of Red 1. This strength is in spite of Red's intrusion on move 5. (I first saw this move in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2090091&nmove=16 this bot game] from [https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=739 this thread on LittleGolem].)
#* Variations of this pattern appear frequently in obtuse corner joseki. If Red started with the 5-5 obtuse corner, then the equivalent Blue 4 is [https://hexworld.org/board/#19nc1,e15f13:pg14 also] strong.
#* Another simple example is Red 5 in [https://hexworld.org/board/#19nc1,d16d15f14:pe13 this position].
# ''Inference:'' Let us consider [https://hexworld.org/board/#19nc1,d16c17d17c18d18:p this position] again. Suppose Blue tenukis on move 6. What is a good local response for Red on move 7? There is more than one right answer, but you just need to find one move that isn't a blunder.
#* ''Hint:'' Use the two patterns above to come up with an answer.
#* ''Hint:'' Consider that Blue has a 3rd row ladder escape, so Blue d15 is locally strong but Blue e14 is weak. This is the case for future blue stones in this local region, even though Blue already played elsewhere on move 6. Therefore, Red should perhaps try to play a move that makes Blue d15 less attractive, even if it comes at the concession of making Blue e14 more attractive (since the latter is ''a priori'' a weak move, such a concession gives up less than it otherwise might).
#* ''Answer:'' Red 7 [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13 here] is a good local move.
#** Why? Ignoring stones 2-5, the presence of Red 1 and 7 makes Blue e14 more attractive. Red's 4-4 stone, much like the obtuse corner opening, tends to shift Blue's efficient stone locations up one row. With that in mind, Red's move 7 is a bit like Red e14 on an empty board, where Blue e15 is [https://hexworld.org/board/#19nc1,e14e15 the best response]. However, stones 2-5 make Blue e14 a bad move, despite Red 7 making it slightly less bad (this is the concession-but-not-really-a-concession that Red intended).
#** Let's now consider Blue d15. Normally a decent local move after stones 1-5, it becomes quite weak after Red 7, because of [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13d15f14 this sequence]. This is a typical example of the second pattern in this extended example. (Actually, if Blue had two moves in a row in this region, Blue 8 wouldn't be so bad because she could follow up with f14, where Red wants to play move 9. This is a tactical situation where it's bad for Red to tenuki.)
#** If we wanted to be complete, Blue actually has another option for move 8, f14 first which threatens d15: [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13f14d15d14e14 here]. This is an example of the "skew cut" from above. Red should respond with d15 himself, or else Blue can play d15. This is probably Blue's best local response.
#** Every move makes a concession, but some concessions are more effective than others. As alluded to in the hint, Red 7 weakens a strong Blue reply (d15) while strengthening a weak Blue reply (e14). Red's goal is to minimize the strength of Blue's ''best'' response (rather than the strength of Blue's average response), and Red 7 at e13 accomplishes that beautifully.

Part of getting stronger at Hex (as with other games) is being able to compress knowledge and patterns effectively. If you understand this extended example, you can infer the best move in many similar but novel situations, without memorizing each one individually. If you forget which move is exactly the best, you can recover the right move with high probability by just reasoning about related patterns that you do know are strong or weak. The "A3 escape trick" example earlier in this article is another instance of this concept.

On larger boards and particularly near the opening or middlegame, when you have a lot of time to think about a position, it may be worth spending a good chunk of it making inferences and scoping out moves that are either good or to be avoided by each side, rather than spending most of your time calculating tons of lines and variations.
* In fact, once you've scoped out that "moves A and B are good for Red, while moves X and Y are bad for Red" and similarly for Blue, that makes the process of analyzing and pruning lines much more efficient.

==Joseki==

* The 4-3 corner move is almost always a mistake compared to the 5-4, even on 11×11. If your opponent plays 4-3, 90% of the time the best local reply is 3-3. If 4-3 is played, here are some common sequences:
: <hexboard size="5x7" float="inline" coords="hide" edges="bottom right" contents="R 1:e2 B 2:e3 R 3:d3"
/> <hexboard size="5x7" float="inline" coords="hide" edges="bottom right" contents="R 1:e2 B 2:e3 R 3:f2 B 4:f3 R 5:d3 E a:b4 b:c5"
/>
: In the second joseki, Blue should play either (a) or (b), or she can defer the question until later. On larger boards with no nearby stones, (b) is often better.
* If your opponent plays the 5-4 joseki, a reasonable choice is to always play the 4-4 "high intrusion." KataHex prefers the 4-4 response by far, especially on 13×13 and larger. It's easily the "safest" choice, and many strong players have recently shown a clear preference for 4-4.
* In this often-played joseki, Red 7 is a mistake because of the territory Blue gets after move 8:
: <hexboard size="6x10" float="inline" edges="bottom right" coords="none" contents="R 1:g2 B 2:g3 R 3:h2 B 4:h4 R 5:i3 B 6:i4 R 7:f3 B 8:e5 R 9:d4 B 10:c6 R 11:b5 B 12:c4 R 13:c5" />
:* If the obtuse corner on Red's same edge is empty, a more principled response for Red is to play in that corner, like this:
:: <hexboard size="6x14" float="inline" edges="bottom left right" coords="none" contents="R 1:k2 B 2:k3 R 3:l2 B 4:l4 R 5:d3" />
:* If the obtuse corner is already occupied by Blue, then Red should still respond in the obtuse corner. One reasonable option (among others) is this:
:: <hexboard size="6x14" float="inline" edges="bottom left right" coords="none" contents="B d3 R 1:k2 B 2:k3 R 3:l2 B 4:l4 R 5:e3" />
* See more discussion of acute corner joseki at [[Openings_on_19_x_19#Acute_corner_theory]]. For the obtuse corner, see [[Openings_on_19_x_19#Obtuse_corner_theory]] which covers responses to the 4-4 and 5-5 obtuse corner in great detail. Most of this is applicable to 13×13 and larger, and not just 19×19.

==Specific tactics==
===Threatening a bridge cut===
One of my favorite tactics is threatening a bridge cut, where the opponent needs to play too close to their own edge to defend:
<hexboard size="8x8"
float="inline"
edges="bottom left"
coords="hide"
contents="B *:d5 g2 R 1:f4 B 2:e3"
/>
Red 1 is a strong move, because it forces Blue to respond at 2 (or else Red will play the bridge cut there), but Blue 2 is normally overplaying on the left side when combined with Blue's stone at (*). Be on the lookout for this when your opponent plays the 4-4 and 7-7 obtuse corner, or 5-5 and 8-8.

This tactic can work well in other configurations, like [https://hexworld.org/board/#19c1,:pg8:pg11h9 this] and [https://hexworld.org/board/#19c1,:pe9:ph9g8 this]. However, be careful when using this tactic! When Blue isn't actually making a concession by bridging her two stones, like when the pattern occurs further from her own edge, it can be a blunder for Red to threaten a bridge cut. [https://hexworld.org/board/#19nc1,:pg8:pj5i7h6 Here] is an example where Red 5 is a blunder in the absence of nearby stones.

[https://www.littlegolem.net/jsp/game/game.jsp?gid=2436000&nmove=12 Here] is an example from a real game. Black's best move is 13. h5 (in the actual game, Black doesn't play this until move 19 but is still arguably rewarded for doing so). One might think that h5 is too strong near the top edge because of c2 and b5, but this is not the case because the white stone on c5 partially neutralizes b5.

[[category:strategy]]

User:Hexanna

2025-12-13T13:28:48Z

Hexanna: compress spacing

User:Hexanna

2025-12-12T05:36:29Z

Hexanna: remove unnecessary move numbers

User:Hexanna

2025-11-30T17:59:51Z

Hexanna: typo

User:Hexanna

2025-11-30T04:48:47Z

Hexanna: grammar

User:Hexanna

2025-11-30T04:22:00Z

Hexanna: advanced strategy: leaving the question open (part 3)

User:Hexanna

2025-11-30T03:42:47Z

Hexanna: advanced strategy: leaving the question open (part 2)

User:Hexanna

2025-11-30T01:43:20Z

Hexanna: advanced strategy: leaving the question open (part 1?)

User:Hexanna

2025-11-29T23:13:35Z

Hexanna: small bit on ladders

Strategic advice from KataHex

2025-11-29T22:41:09Z

Hexanna: remove a useless pattern; grammar

==Introduction==

Several hex bots have popped up in the last few years. Many are neural nets and adaptations of Go bots like AlphaGo Zero. The most recent (and likely strongest) as of 2025 is [[KataHex]], which is based on the open-source KataGo. These bots have influenced human play in many aspects already; one of the most notable is the bots' preference for the 4-4 obtuse corner over the 5-5. Many old strategy guides have become partially (if not mostly) outdated.

This article attempts to bring some strategy advice "up-to-date". There's a lot of random insights; most are a direct result of playing around with KataHex evaluations for thousands of hours, including analyzing the games of strong players for common mistakes. Since this is strategic advice and not tactics, the advice won't apply 100% of the time, and many players may disagree with some parts. Win percentages and evaluations in this article are based on KataHex net [https://github.com/hzyhhzy/KataGo/releases/tag/Hex_20240812 20240812], which is significantly stronger than previous nets.

Care has been taken to only draw conclusions when there are multiple examples of KataHex preferring something. However, the main goal is to give actionable advice to make players stronger, and there isn't rigorous proof for basically any of these statements.

===Big picture, and some meta-points===

* '''TL;DR:''' What are the most important principles? Even if you learn nothing else, you should learn these:
*# Avoid playing close to your own edge, especially in the opening, unless you have a very good reason to do so.
*#* Think of your own edge as a whole row of stones of your color. You already have a bunch of stones (and thus strength) there, and playing yet another one close by would be a waste. That stone is much better served elsewhere, where you don't have as much strength.
*# Bridges are the most efficient way to connect your own stones—not adjacent stones, trapezoids, crescents, or hammocks. If there are no opposing stones nearby, playing adjacent to an existing stone of your color is usually a mistake; if you want to do that, consider whether there's a better move a bridge away from that stone.
*#* If there are nearby opposing stones, things become more nuanced, but bridging to your existing stones is often still a good idea.
*# Usually, cut through an opponent's bridge if given the opportunity, and avoid letting it happen to you. In the pattern below, Red cuts through Blue's two stones a bridge apart. This is very good for Red and bad for Blue, because it's very hard for Blue to connect the two stones now.
*#* <hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 c1 R b1 1:b2"
/>
* This guide assumes you've read [http://www.mseymour.ca/hex_book/hexstrat.html ''Hex: A Strategy Guide''] or have equivalent knowledge.
** ''Hex: A Strategy Guide'' predates KataHex, but it's very well made, and 90% of the advice is still good.
* On smaller boards, tactics and raw calculation matter more. On larger boards, strategy matters more. (It's ultimately a spectrum, and it varies by skill level: for beginners, size 13 might be considered large; for strong players, tactics can matter a great deal on size 13.)
** On smaller boards, the exact board size affects strategy (some moves can be good on 11×11 but not 12×12). The larger the board, the less the exact board size matters. The general strategy for 19×19 and 20×20 should be nearly identical.
** On smaller boards, there is more space close to corners and edges. On larger boards, there is more "open" space (hexes not close to corners and edges). I intentionally said "open" space and not "center" space, because there is not much special about hexes near the center of the board, except that they are far away from corners and edges. But on large boards, there are many such areas.
** In particular, the larger the board size (starting at around size 13–15), the more important it is to know how to play in "open" areas. Many existing guides cover corner/edge strategy heavily—and correctly so, as corners/edges remain important up to very large sizes (25–30 or so). People read those guides and often know how to play the first 10–20 moves of a 17×17 game, but often not after that, when most of the game is still remaining!
** Strategy can also be ''local'' or ''global''. Local strategy tells you how to play within a small region of the board, and global strategy tells you how to play in relation to the whole board (or across multiple local regions). Both are important.
** The strategy of "open" areas is mostly local, and so the larger the board, the higher the relative importance of local strategy. Why?
*** On a large Hex board, the "middle" is relatively homogeneous—two separate open areas near the middle are basically the same in terms of how you should play them; it doesn't matter that one of the areas is 15 hexes away from an edge and the other one 18. Because of this homogeneity, the same basic principles will apply multiple times in the span of a game.
*** Knowing how to play across multiple local regions is very hard, and it's often different across games (especially for humans). It's better to master local play first, since similar local play principles will be relevant every game.
* How should a game of Hex (on larger boards) broadly feel like?
** While the win condition is binary (you either connect or you don't), the actual gameplay feels like a gradual accumulation of tiny advantages and disadvantages. Strong players often "know" they're winning in the middlegame, not because they see an actual way to connect, but because their opponent made some mistakes, however small, along the way. (You rarely know for sure who's winning until the end, and you're ultimately making an educated guess.)
** In particular, it's usually very hard to tell who's winning just by looking at a position. Contrast that with games like chess, where you can often tell by counting up material and broadly looking at positional factors. In Hex, it's much more tractable to tell how good your position is ''relative to a few moves ago.''
** Relatedly, there's (almost) no meaningful notion of "long-term strategy" in Hex. If anything, it can be harmful to create a long-term plan and stubbornly stick to it, because the moves you should play often differ greatly depending on your opponent's responses.
** Hex strategy is equally what to play and what ''not'' to play. The latter is frequently underemphasized. Each stone is worth a lot, and even one bad move can "undo" 10–20 good ones. The risk-reward balance strongly favors avoiding mistakes, rather than attempting risky but potentially "brilliant" moves.
* Now some points about how to learn. From what I observe, people often read strategy advice and then fail to follow it consistently. This is often either because of skepticism, or not realizing just how often particular pieces of advice should be followed (it's tempting to think that you've found the exception to the rule).
** It's healthy to be skeptical—there is a lot of ''bad'' strategy advice, and it would probably do more harm than good to implement every piece of advice you come across. So I'm not expecting readers to take everything in this guide at face value either (and many statements have nuances/exceptions, some of which I don't even realize should be explained to readers; it's hard to translate precise intuitions into words after all). Unfortunately, separating the signal from the noise is hard. If you're skeptical about a principle or how strongly it holds, try running it through KataHex yourself (if you have it installed), and try variations to understand the nuances, when the principle holds, and importantly ''why'' it holds. Sometimes people are skeptical of KataHex itself or think they can do better; I think there are good heuristic reasons why this thinking is misguided (the most simple one is that KataHex is far stronger than the strongest humans).

==General==

===Common mistake: playing near your own edge===

Playing near your own edge with no nearby stones is a bad idea in the opening and even middlegame. This is one of the most common mistakes among human players. You need a strong (often tactical) reason to do otherwise.
* When is playing near your own edge a good idea? There are a few cases, like when you're playing adjacent to existing stones, responding to a joseki, or pushing a ladder as the attacker, but all of these involve stones in the vicinity.
* This holds most strongly for 13×13 to 19×19, but applies more often than not to 11×11 too.
* Playing near your own edge in the opening or early middlegame should be a "last resort". I would recommend never doing so in the first 30-40 moves in 19×19 unless you have a very good reason.

===Common mistake: bad minimaxing===

Many intermediate and strong players are too eager to minimax. They play bad minimaxing moves (that KataHex thinks are inferior to directly connecting) more often than they fail to minimax. The issue is that minimaxing isn't always free; your opponent can often intrude for useful territory, and sometimes the territory your opponent gains is more useful than what you gain by minimaxing. It's a difficult skill to judge which player gains more from a minimax, but there are common patterns.
* Players frequently attempt to minimax by playing adjacent to their existing stone on the 5th or 6th row of their own edge. Playing two adjacent stones like this is not very efficient and typically a mistake.
* Blue's minimaxing attempt is unsuccessful; KataHex already thinks Red is 98% to win: https://hexworld.org/board/#15nc1,c2e11c13e10b12e9c10d8b9c7a8b6c6b7d7c8e8d9
* A common example; Red is 98% to win: https://hexworld.org/board/#13nc1,c2d10c11c10b11c9b8a10. Note that Red 7 at b8 is stronger than at a10, because b8 combines well with c2 and b5 on 13×13. On 15×15, KataHex prefers 7. a12 (the equivalent of a10 on 13×13) instead.
* A case where minimaxing is in fact the best move: https://hexworld.org/board/#13nc1,c2d10c9d11

===Common mistake: bad bridge peep===

A common scenario is the [[Peep#Bridge_peep | bridge peep]]. Blue should play at (*) as long as she thinks it's forcing enough for Red to defend his bridge.

However, even if Blue hasn't played (*) yet, Red should usually mentally place a blue stone there. Especially on larger boards, Blue will be able to play at (*) sometime in the future. This is important: even if (*) is empty, I like to think of this pattern as ''implying'' a blue stone at (*) (in the absence of tactics that somehow prevent Blue from playing there). Another way (useful for me, but the analogy might be nonsense to others) to think about it is that Blue's stone at A is well-placed enough relative to Red's bridge that she has some "potential energy" at (*).

<hexboard size="3x3"
coords="hide"
edges="hide"
contents="R c1 b3 B A:a2 E *:c2"
/>

However, 80% of the time, if there's no blue stone at A, Blue should not intrude on either side of the bridge. Intruding is usually a mistake because it settles the [[question]] for Red. If Red knows he will have a stone at A or B because Blue already intruded in the other spot, Red can better plan his future moves because of that! There are exceptions; if Blue is confident that only one of the intrusions will ever be useful for her, then intruding is often strong.

<hexboard size="3x2"
coords="hide"
edges="hide"
contents="R b1 a3 E A:a2 B:b2"
/>

===Ladders===
* Don't worry about switchbacks, climbing, or ladder creation templates on 15x15 or larger. They can be theoretically interesting and make for good puzzles, but games are rarely close enough for these to matter. I personally don't think about them on any board size, but others may reasonably disagree with this approach.
* Q: When should the defender of a ladder yield? A: Usually, don't yield unless your opponent can cut through two of your stones, one above and one below the ladder. In that case, consider yielding.
** [https://hexworld.org/board/#13nc1,c2e9c10d8b9c11d10d12 Here] is an example. If Blue instead plays 8. d11, Red 9. e10 cuts through e9 and d11.
** In [https://playhex.org/games/66b8a125-bf02-42d2-8900-af3042a8ccf8 this game] from the [[Tournaments#Hex_Monthly | Hex Monthly 21]] ([https://hexworld.org/board/#14c1,c2d11k10k4b5f10g9i9i8g10h9h10j8j9b12c10l8k8l7k7l6k11j11h12m5k6l4m4l5l9k9h13i13l3k5i5j5j4i6h6f8h7i7f9g8f7e8e7d8d7b8c8b9c9a11b13d10c11c12,c13d12d13e11:rw HexWorld link]), the yielding move 60. c14 is winning, while the played 60. c13 (which allows Red d12, cutting through Blue's c13 and d11 stones) is losing.

===Other===

* Be aggressive; always be ready to capitalize on an opponent's mistake. Here are a few illustrative examples of this and some of the above concepts:
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2368410&nmove=8 this game], 6. c9 was a mistake (and an example of incorrect minimaxing); 7. a10 is the [[Openings_on_19_x_19#4-4_obtuse_corner | standard reply]]. White should have played 8. b8; j4 was a mistake. Black's 9. b8 is the "capitalizing" move; KataHex already says 99% win rate for Black. In this case, there were several good options on move 9. These alternatives are threatening enough that Black didn't have to play b8, but this won't always be true, like in the next example.
** [https://littlegolem.net/jsp/game/game.jsp?gid=2362913&nmove=26 Here] is another example (interestingly, White resigned in a winning position according to the bot, though the final position is highly tactical). KataHex actually thinks Black has only an 35% win rate after 26. m6. However, 27. d11 is a "minimaxing with adjacent stones" mistake, bringing Black's win rate down to 3%.
** [https://www.littlegolem.net/jsp/game/game.jsp?gid=2400947&nmove=6 Here] is an example where White's win rate increased consistently from 65% after move 6 to 98% after move 16. The aggressive bridging sequence in moves 8–16 is very strong because it basically neutralizes Black's stones on d16 and d17; it is a big mistake for Black to ignore the threat here. Move 11 is also an example of a bad bridge peep.

* Don't play a move that makes your opponent's existing stones unnecessarily well-placed relative to your new stone. This is covered briefly in a couple places in the [[Openings_on_19_x_19#General_principles | 19×19 opening guide]]. Here are a couple more examples.
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2369954&nmove=26 this game] ([https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6 HexWorld link]), what should Black play on move 27? White is threatening to connect 26 back to the 16–24 group if Black isn't careful. Black's obvious options are to play at c8 and c9. Which option is better?
** A tempting answer is c8 because it appears to connect more closely to the top. However, note that White's stone on move 14 functions exactly like an opening a3 stone. For example, it can escape 2nd but not 3rd row ladders. If Black plays c8, White can play the [[A3 escape trick]], as shown in [https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6c8e7d8e8d9e9 this continuation]. This sequence allows White to "make the most" of stone 14, and Black shouldn't allow it. Better for Black is 27. c9, which doesn't allow White to carry out a 3rd-to-5th row switchback. Indeed, KataHex evaluates c9 as 97% to win for Black and c8 as only 30%.
** Consider [https://hexworld.org/board/#13n,c2d10c11c10b11e6b10d8l4l3c8d5b5d6h7h6j5i6k7 this position] (which is based on [https://www.littlegolem.net/jsp/game/game.jsp?gid=2369782 this game]). Where should White play in the lower-right acute corner? Black had just played at the "3-7" point from White's perspective; as mentioned in the 19×19 opening guide, a strong response to 3-7 is 4-4 (at j10) instead of the usual 5-4 (at j9). A player who is short on time can play j10 without thinking too hard. As it turns out, KataHex thinks j10 is the best move, evaluating it as 91% for White, whereas j9 is only around 50%.

* Sometimes, you know you have a completely winning position after your opponent blunders very early in the game. Your goal is to preserve that win until the end of the game. Don't overextend yourself and try to win too "quickly", possibly making possibly suboptimal moves in the process. Even when KataHex plays itself with a significant handicap (like playing without swap), a significant fraction of the board seems necessary to carry out the win, assuming strong defense. I prefer taking my time to "fill the board" by focusing on playing moves at least as good as my opponent's, instead of worrying about connecting in as few moves as possible.

==Local patterns==

An especially effective way to improve at Hex is to learn and gain intuition about lots of small local patterns. These patterns are small enough that they occur basically everywhere when you play Hex. Knowing the patterns can help you avoid blunders (for example, avoiding adjacent or bridge ''allow-cuts'', as explained below) and make good moves when it's otherwise unclear what to play.

===Cuts===
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 c1 R b1 1:b2"
/>
<hexboard size="1x3"
float="inline"
edges="none"
coords="none"
contents="B a1 c1 R 1:b1"
/>
<hexboard size="3x4"
float="inline"
edges="none"
coords="none"
visible="-(a1 b1 d1 a3 c3 d3)"
contents="B a2 d2 R c1 1:b3 S b2 c2 E *:(b2 c2)"
/>
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a3 c1)"
contents="B a1 c3 R 1:b2 S b1 a2 c2 b3"
/>
Consider the four patterns above. Let's call the patterns '''adjacent cut''', '''near cut''', '''short bridge cut''' (or just '''bridge cut'''), and '''long bridge cut''' respectively. In some sense, Red is "cutting" through two of Blue's stones in each pattern, so Red is the '''attacker''' and Blue is the '''defender'''. We can also use '''cut''' as an attacking verb and '''allow-cut''' as a defending verb. For example, the adjacent cut pattern is favorable for the attacker (Red above), so we can say that it's a good idea to ''adjacent-cut'' through opponent stones, and a bad idea to ''adjacent allow-cut'' (meaning it's a bad idea to play a move that allows your opponent to adjacent-cut through two of your stones).

Why is it strong to adjacent-cut (and equivalently, why is it a mistake to adjacent allow-cut)? The most intuitive reason is that if Blue allow-cuts, it takes 3 extra moves for her to connect her two stones, as shown below. In a sense, this makes it unlikely for both stones to be concurrently useful in the final connection — if one of the stones proves useful, chances are the other one won't be.
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c3)"
contents="B a2 c1 1:a3 2:b3 3:c2 R b1 b2"
/>

The (short) bridge cut is strong for the attacker (Red above) because he captures the two hexes marked (*). If Blue allow-cuts, often the damage has already been done even if Red doesn't actually cut through immediately on the next turn. (However, note that this isn't always true; some tactical situations require Red to cut through immediately instead of playing elsewhere.)
* There are situations where Blue ''should'' allow-cut. If Red's cutting move comes with concessions, like the move is on his second or third row (and especially if Red already has a stone near his own edge, so cutting through would be overplaying that edge), the pattern is much less favorable to Red.

The near cut and long bridge cut are also favorable for the attacker, but much less clearly so — there are many more "exceptions" where cutting is bad and allow-cutting is fine. On the other hand, the adjacent and short bridge cuts are quite strong, and cutting through is usually the best move (unless doing so involves playing a stone that's too close to your own edge, or another major concession).

Note that allow-cuts can occur not only ''proactively'' before the attacker plays the cutting move, but also ''retroactively'' after the "cutting" move is played (of course, it's not really a "cutting" move until after the defender allow-cuts). In other words, Blue 1 is considered an allow-cut in both of these examples (proactive on the left and retroactive on the right), and they are equally bad for Blue assuming that Red intends to cut in the proactive case:
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 1:c1 R b1"
/>
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 1:c1 R b1 b2"
/>

An important way of pruning moves and avoiding blunders is making sure you don't play moves that adjacent or bridge allow-cut, either proactively or retroactively, unless you have a good reason to do so. In particular, the retroactive case can be easy to overlook if you're not used to thinking in this way.

===Small patterns===
<hexboard size="2x2"
float="inline"
edges="none"
coords="none"
visible="-a1"
contents="B b2 R b1 1:a2"
/>
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 a3 c2 c3)"
contents="B b3 R c1 1:a2"
/>
<hexboard size="4x2"
float="inline"
edges="none"
coords="none"
visible="-(a1 b4)"
contents="B a4 R 1:b1"
/>

In each of these patterns, we will discuss the strength of Red's move 1 ''in a vacuum'', meaning in the absence of nearby stones, like near the middle of an empty board on 19×19 or even larger boards. It is most useful to analyze each pattern without the influence of nearby stones or edges (which can be thought of as virtual rows of stones).
* Pattern 1: It's typically an inaccuracy for Blue to play adjacent to a red stone in a vacuum. Red's best response is usually playing adjacent to both stones, as in Red 1.
* Pattern 2: Blue plays a bridge away from Red's stone. This is often a good move, and Red 1 a bridge away from both stones is also good.
* Pattern 3: Red 1 is a classic block. This is often fine for Red, and Blue doesn't have an obvious best local response. Blue could tenuki here.

===Capture patterns===

Know the most common [[captured cell]] patterns. It really reduces the number of moves you have to consider.

The bridge cut is one of the most important capture patterns.

* A typical example can be found in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2233736&nmove=10 this game]. After 10. g11, Black would have the easiest time with 11. h9, with 95% win rate, instead of the played k8 with 74% win rate. After k8 was played, one of White's best moves would've been h9 to prevent Black from playing the same. KataHex does think Black has other reasonable options on move 11, but they mostly appear to just delay h9.

Here is another common capture pattern. The other patterns on [[captured cell]] are useless because they have too many Red stones adjacent to each other, which is too inefficient to occur in normal play.

<hexboard size="3x3"
coords="hide"
edges="hide"
contents="R c1 c3 b3 B a2 S b2 c2 E *:(b2 c2)"
/>

This ''one-sided'' capture pattern is very common. I like to mentally replace A with a blue stone and (*) with a red stone (even though Blue won't always achieve this), because Blue should never play at (*). [[Edge template IV3a]] is useful to know here.
<hexboard size="3x2"
coords="hide"
edges="hide"
visible="-a1"
contents="R b1 a3 b3 S a2 b2 E *:b2 A:a2"
/>

The trapezoid and crescent also one-sided-capture the hexes at (*). Anecdotally, this fact is more useful on smaller boards like 11×11, but it's less common overall than the previous capture patterns.
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
visible="-a1"
contents="R b1 a3 c3 c1 S a2--c2 b3 E *:(c2 b3)"
/>
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
visible="-a1"
contents="R a2 b1 c1 c3 S b2 c2 a3 b3 E *:c2"
/>

==How to think about the opening and middlegame==

Some of these points sound like hyperbole, but they are a fairly accurate representation of how I think about Hex, especially on larger boards. There are exceptions, but exceptions are rarer than one might think.

* In the first half of the game, focus all of your attention into playing "efficient" moves. These are moves that work well with existing stones and board edges. Excellent intuition of efficient patterns is enough to get strong middlegame positions against even top players (as of 2024).
* Don't worry about tactics until the late middlegame. It's a waste of time to analyze sequences >10 moves deep. In the opening, the best move you can find with only intuition is no worse than the best move you can find with calculation.
** Why? Compared to bots, humans are very bad at calculation. Your intuition (which you learn directly from KataHex) is more useful than hypothetical hours or days of error-prone human analysis.
* Use your knowledge of efficient patterns to plan ahead. This is where you want to do shallow analysis, typically 1-5 moves deep. A good move does as much as possible:
** allows you to play efficient moves later
** prevents your opponent from playing an efficient move outright
** allows your opponent an efficient move, but only with concessions (say, a locally efficient but globally inefficient move, or a move that's efficient with relative to one stone but inefficient relative to another)

These points require a lot of elaboration. What do I mean by "efficient"? (Not to be confused with [[efficiency]].) There are two broad notions, '''global''' and '''local''' efficiency.
* Globally efficient stone: Broadly, any move that isn't too close to your own edge. Moves that are strong in the absence of nearby stones (like on an empty board). Corner moves are very efficient. On 19×19, moves near your opponent's 5th row are very efficient.
* Locally efficient stone: In the presence of nearby stones, some configurations are broadly "good" or "bad" for one player. They aren't ''always'' good or bad because of tactics, but because humans can't calculate everything, it's extremely useful to have decent heuristics that work most of the time.

In a sense, you want to play the move that maximizes the "sum" of local and global efficiency. This isn't an exact science; use your intuition to make an educated guess.
* The best move locally is rarely the best move globally. Hex is a game of concessions and tradeoffs. There is no "free lunch," unless you count concepts like inferiority/domination.
* Conversely, if it looks like your opponent is beating you on one side of the board, there are two possibilities:
*# You played suboptimal moves in that region.
*# Your opponent overplayed that region, which necessarily means they made concessions elsewhere (usually the opposite side of the board). This means your opponent ''underplayed'' the other region.
* If option 1 is true, you are probably losing anyways, so assume (or hope) option 2 is true. If you opponent overplays region A and underplays region B, it's typically a good idea to focus your play in region B. Why? If you keep playing in region A, you're potentially helping your opponent out of their mistake, by making your opponent less over-connected in region A. For example, your opponent could have a minimaxing move that makes them much stronger in region B, at the "concession" of making them weaker in region A — but this isn't really a concession because your opponent had strength to spare in region A.

The discussion so far has been highly abstract. What does efficient play look like in practice? As explained earlier, it's strong to adjacent-cut and bridge-cut through your opponent's stones. The small patterns in the "Capture patterns" section above are also locally efficient for the capturing player. Of course, just as offense and defense are equivalent in Hex, so are playing locally efficient moves and inducing your opponent to play locally ''inefficient'' moves.

In the following examples, P1 and P2 are efficient patterns for Red relative the corresponding edges, while P3 is inefficient. P1 is common when Red opens with a first column opening. P2 requires elaboration: normally Red 1 is a weak move because it's too close to Red's own edge. However, when Red must play on his first 3 rows (because of very limited space), Red 1 or its mirror image at (*) is usually the best, because it prevents Blue from playing there and forming the equivalent of P1 herself. P2 also occurs in theoretical contexts, like [[Template Va#If Blue moves at y:]]. In P3, Red 1 is a tempting (because of [[edge template III1b]]), common, but inefficient move, because Blue's response at one of (*) is usually strong. If Red must play near his own edge, he should instead play as in P2 above.
P1: <hexboard size="3x2"
float="inline"
edges="left"
coords="none"
contents="R a2 1:b3"
/>
P2: <hexboard size="2x6"
float="inline"
edges="bottom"
coords="none"
contents="B c2 R 1:e1 E *:b1"
/>
P3: <hexboard size="3x5"
float="inline"
edges="bottom"
coords="none"
contents="B c3 R 1:d1 E *:(b2 e2)"
/>

===Making inferences: An extended example===

Use the patterns and intuitions that you know, to reason and make inferences in unfamiliar positions. You might not always find the best move, but you can often prune bad moves. Here is an extended example, involving several closely related patterns that are useful to know in isolation.
# ''Pattern:'' If you play the 4-4 3-3 obtuse corner joseki, it is important to know that Blue 6 [https://hexworld.org/board/#19nc1,d16c17d17c18d18e14c15 here] is weak.
#* This is because Red 7 undermines Blue 6 and also reduces Blue 2/4 from a 3rd row ladder escape to a 2nd row escape. Also, Blue's potential intrusion at c16 doesn't gain her anything.
#* Instead, [https://hexworld.org/board/#19nc1,d16c17d17c18d18d15 here] is a good local response for Blue. This gives Blue a 5th row ladder escape while blocking Red 1.
#* On the other hand, if Blue only has a 2nd row escape and doesn't have the "potential energy" to get a 3rd row escape, you might guess that Blue 6 above becomes a good local move, because Red 7 loses one of its main strengths. Such a guess would be correct: in [https://hexworld.org/board/#19nc1,d16c17p15d17e16g16e17f18d18e14c15 this position], Blue 10 is a strong move. On the other hand, Blue 10 at d15 would be weak here, as it's quite wasteful, only converting a 2nd row escape into a 3rd row escape.
# ''Pattern:'' If Red tenukis on move 3, then Blue 4 is locally strong in [https://hexworld.org/board/#19nc1,d16e14:pf15 this position]. In particular, Blue 4 is much stronger than bridging directly towards Blue's own edge with g13, because the former is a strong blocking move that reduces the effectiveness of Red 1. This strength is in spite of Red's intrusion on move 5. (I first saw this move in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2090091&nmove=16 this bot game] from [https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=739 this thread on LittleGolem].)
#* Variations of this pattern appear frequently in obtuse corner joseki. If Red started with the 5-5 obtuse corner, then the equivalent Blue 4 is [https://hexworld.org/board/#19nc1,e15f13:pg14 also] strong.
#* Another simple example is Red 5 in [https://hexworld.org/board/#19nc1,d16d15f14:pe13 this position].
# ''Inference:'' Let us consider [https://hexworld.org/board/#19nc1,d16c17d17c18d18:p this position] again. Suppose Blue tenukis on move 6. What is a good local response for Red on move 7? There is more than one right answer, but you just need to find one move that isn't a blunder.
#* ''Hint:'' Use the two patterns above to come up with an answer.
#* ''Hint:'' Consider that Blue has a 3rd row ladder escape, so Blue d15 is locally strong but Blue e14 is weak. This is the case for future blue stones in this local region, even though Blue already played elsewhere on move 6. Therefore, Red should perhaps try to play a move that makes Blue d15 less attractive, even if it comes at the concession of making Blue e14 more attractive (since the latter is ''a priori'' a weak move, such a concession gives up less than it otherwise might).
#* ''Answer:'' Red 7 [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13 here] is a good local move.
#** Why? Ignoring stones 2-5, the presence of Red 1 and 7 makes Blue e14 more attractive. Red's 4-4 stone, much like the obtuse corner opening, tends to shift Blue's efficient stone locations up one row. With that in mind, Red's move 7 is a bit like Red e14 on an empty board, where Blue e15 is [https://hexworld.org/board/#19nc1,e14e15 the best response]. However, stones 2-5 make Blue e14 a bad move, despite Red 7 making it slightly less bad (this is the concession-but-not-really-a-concession that Red intended).
#** Let's now consider Blue d15. Normally a decent local move after stones 1-5, it becomes quite weak after Red 7, because of [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13d15f14 this sequence]. This is a typical example of the second pattern in this extended example. (Actually, if Blue had two moves in a row in this region, Blue 8 wouldn't be so bad because she could follow up with f14, where Red wants to play move 9. This is a tactical situation where it's bad for Red to tenuki.)
#** If we wanted to be complete, Blue actually has another option for move 8, f14 first which threatens d15: [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13f14d15d14e14 here]. This is an example of the "skew cut" from above. Red should respond with d15 himself, or else Blue can play d15. This is probably Blue's best local response.
#** Every move makes a concession, but some concessions are more effective than others. As alluded to in the hint, Red 7 weakens a strong Blue reply (d15) while strengthening a weak Blue reply (e14). Red's goal is to minimize the strength of Blue's ''best'' response (rather than the strength of Blue's average response), and Red 7 at e13 accomplishes that beautifully.

Part of getting stronger at Hex (as with other games) is being able to compress knowledge and patterns effectively. If you understand this extended example, you can infer the best move in many similar but novel situations, without memorizing each one individually. If you forget which move is exactly the best, you can recover the right move with high probability by just reasoning about related patterns that you do know are strong or weak. The "A3 escape trick" example earlier in this article is another instance of this concept.

On larger boards and particularly near the opening or middlegame, when you have a lot of time to think about a position, it may be worth spending a good chunk of it making inferences and scoping out moves that are either good or to be avoided by each side, rather than spending most of your time calculating tons of lines and variations.
* In fact, once you've scoped out that "moves A and B are good for Red, while moves X and Y are bad for Red" and similarly for Blue, that makes the process of analyzing and pruning lines much more efficient.

==Joseki==

* The 4-3 corner move is almost always a mistake compared to the 5-4, even on 11×11. If your opponent plays 4-3, 90% of the time the best local reply is 3-3. If 4-3 is played, here are some common sequences:
<hexboard size="5x7"
coords="hide"
edges="bottom right"
contents="R 1:e2 B 2:e3 R 3:d3"
/>

<hexboard size="5x7"
coords="hide"
edges="bottom right"
contents="R 1:e2 B 2:e3 R 3:f2 B 4:f3 R 5:d3 E a:b4 b:c5"
/>

In the second joseki, Blue should play either (a) or (b), or she can defer the question until later. On larger boards with no nearby stones, (b) is often better.

* If your opponent plays the 5-4 joseki, a reasonable choice is to always play the 4-4 "high intrusion." KataHex prefers the 4-4 response by far, especially on 13×13 and larger. It's easily the "safest" choice, and many strong players have recently shown a clear preference for 4-4.

* In [https://hexworld.org/board/#14nc1,k10k11l10l12m11m12j11i13h12g14f13g12g13 this often-played joseki], Red 7 is a mistake because of the territory Blue gets after move 8.
** If the obtuse corner on Red's same edge is empty, a more principled response for Red is to play in that corner, like [https://hexworld.org/board/#14c1,k10k11l10l12d11 this].
** If the obtuse corner is already occupied by Blue, then Red should still respond in the obtuse corner. One reasonable option (among others) is [https://hexworld.org/board/#14c1,:pd11k10k11l10l12e11 this].

* See more discussion of acute corner joseki at [[Openings_on_19_x_19#Acute_corner_theory]]. For the obtuse corner, see [[Openings_on_19_x_19#Obtuse_corner_theory]] which covers responses to the 4-4 and 5-5 obtuse corner in great detail. Most of this is applicable to 13×13 and larger, and not just 19×19.

==Specific tactics==
===Threatening a bridge cut===
One of my favorite tactics is threatening a bridge cut, where the opponent needs to play too close to their own edge to defend:
<hexboard size="8x8"
float="inline"
edges="bottom left"
coords="hide"
contents="B *:d5 g2 R 1:f4 B 2:e3"
/>
Red 1 is a strong move, because it forces Blue to respond at 2 (or else Red will play the bridge cut there), but Blue 2 is normally overplaying on the left side when combined with Blue's stone at (*). Be on the lookout for this when your opponent plays the 4-4 and 7-7 obtuse corner, or 5-5 and 8-8.

This tactic can work well in other configurations, like [https://hexworld.org/board/#19c1,:pg8:pg11h9 this] and [https://hexworld.org/board/#19c1,:pe9:ph9g8 this]. However, be careful when using this tactic! When Blue isn't actually making a concession by bridging her two stones, like when the pattern occurs further from her own edge, it can be a blunder for Red to threaten a bridge cut. [https://hexworld.org/board/#19nc1,:pg8:pj5i7h6 Here] is an example where Red 5 is a blunder in the absence of nearby stones.

[https://www.littlegolem.net/jsp/game/game.jsp?gid=2436000&nmove=12 Here] is an example from a real game. Black's best move is 13. h5 (in the actual game, Black doesn't play this until move 19 but is still arguably rewarded for doing so). One might think that h5 is too strong near the top edge because of c2 and b5, but this is not the case because the white stone on c5 partially neutralizes b5.

[[category:strategy]]

User:Hexanna

2025-11-27T13:41:31Z

Hexanna:

Strategic advice from KataHex

2025-11-27T13:07:00Z

Hexanna: /* Cuts */ small copyedit

==Introduction==

Several hex bots have popped up in the last few years. Many are neural nets and adaptations of Go bots like AlphaGo Zero. The most recent (and likely strongest) as of 2025 is [[KataHex]], which is based on the open-source KataGo. These bots have influenced human play in many aspects already; one of the most notable is the bots' preference for the 4-4 obtuse corner over the 5-5. Many old strategy guides have become partially (if not mostly) outdated.

This article attempts to bring some strategy advice "up-to-date". There's a lot of random insights; most are a direct result of playing around with KataHex evaluations for thousands of hours, including analyzing the games of strong players for common mistakes. Since this is strategic advice and not tactics, the advice won't apply 100% of the time, and many players may disagree with some parts. Win percentages and evaluations in this article are based on KataHex net [https://github.com/hzyhhzy/KataGo/releases/tag/Hex_20240812 20240812], which is significantly stronger than previous nets.

Care has been taken to only draw conclusions when there are multiple examples of KataHex preferring something. However, the main goal is to give actionable advice to make players stronger, and there isn't rigorous proof for basically any of these statements.

===Big picture, and some meta-points===

* '''TL;DR:''' What are the most important principles? Even if you learn nothing else, you should learn these:
*# Avoid playing close to your own edge, especially in the opening, unless you have a very good reason to do so.
*#* Think of your own edge as a whole row of stones of your color. You already have a bunch of stones (and thus strength) there, and playing yet another one close by would be a waste. That stone is much better served elsewhere, where you don't have as much strength.
*# Bridges are the most efficient way to connect your own stones—not adjacent stones, trapezoids, crescents, or hammocks. If there are no opposing stones nearby, playing adjacent to an existing stone of your color is usually a mistake; if you want to do that, consider whether there's a better move a bridge away from that stone.
*#* If there are nearby opposing stones, things become more nuanced, but bridging to your existing stones is often still a good idea.
*# Usually, cut through an opponent's bridge if given the opportunity, and avoid letting it happen to you. In the pattern below, Red cuts through Blue's two stones a bridge apart. This is very good for Red and bad for Blue, because it's very hard for Blue to connect the two stones now.
*#* <hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 c1 R b1 1:b2"
/>
* This guide assumes you've read [http://www.mseymour.ca/hex_book/hexstrat.html ''Hex: A Strategy Guide''] or have equivalent knowledge.
** ''Hex: A Strategy Guide'' predates KataHex, but it's very well made, and 90% of the advice is still good.
* On smaller boards, tactics and raw calculation matter more. On larger boards, strategy matters more. (It's ultimately a spectrum, and it varies by skill level: for beginners, size 13 might be considered large; for strong players, tactics can matter a great deal on size 13.)
** On smaller boards, the exact board size affects strategy (some moves can be good on 11×11 but not 12×12). The larger the board, the less the exact board size matters. The general strategy for 19×19 and 20×20 should be nearly identical.
** On smaller boards, there is more space close to corners and edges. On larger boards, there is more "open" space (hexes not close to corners and edges). I intentionally said "open" space and not "center" space, because there is not much special about hexes near the center of the board, except that they are far away from corners and edges. But on large boards, there are many such areas.
** In particular, the larger the board size (starting at around size 13–15), the more important it is to know how to play in "open" areas. Many existing guides cover corner/edge strategy heavily—and correctly so, as corners/edges remain important up to very large sizes (25–30 or so). People read those guides and often know how to play the first 10–20 moves of a 17×17 game, but often not after that, when most of the game is still remaining!
** Strategy can also be ''local'' or ''global''. Local strategy tells you how to play within a small region of the board, and global strategy tells you how to play in relation to the whole board (or across multiple local regions). Both are important.
** The strategy of "open" areas is mostly local, and so the larger the board, the higher the relative importance of local strategy. Why?
*** On a large Hex board, the "middle" is relatively homogeneous—two separate open areas near the middle are basically the same in terms of how you should play them; it doesn't matter that one of the areas is 15 hexes away from an edge and the other one 18. Because of this homogeneity, the same basic principles will apply multiple times in the span of a game.
*** Knowing how to play across multiple local regions is very hard, and it's often different across games (especially for humans). It's better to master local play first, since similar local play principles will be relevant every game.
* How should a game of Hex (on larger boards) broadly feel like?
** While the win condition is binary (you either connect or you don't), the actual gameplay feels like a gradual accumulation of tiny advantages and disadvantages. Strong players often "know" they're winning in the middlegame, not because they see an actual way to connect, but because their opponent made some mistakes, however small, along the way. (You rarely know for sure who's winning until the end, and you're ultimately making an educated guess.)
** In particular, it's usually very hard to tell who's winning just by looking at a position. Contrast that with games like chess, where you can often tell by counting up material and broadly looking at positional factors. In Hex, it's much tractable to tell how good your position is ''relative to a few moves ago.''
** Relatedly, there's (almost) no meaningful notion of "long-term strategy" in Hex. If anything, it can be harmful to create a long-term plan and stubbornly stick to it, because the moves you should play often differ greatly depending on your opponent's responses.
** Hex strategy is equally what to play and what ''not'' to play. The latter is frequently underemphasized. Each stone is worth a lot, and even one bad move can "undo" 10–20 good ones. The risk-reward balance strongly favors avoiding mistakes, rather than attempting risky but potentially "brilliant" moves.
* Now some points about how to learn. From what I observe, people often read strategy advice and then fail to follow it consistently. This is often either because of skepticism, or not realizing just how often particular pieces of advice should be followed (it's tempting to think that you've found the exception to the rule).
** It's healthy to be skeptical—there is a lot of ''bad'' strategy advice, and it would probably do more harm than good to implement every piece of advice you come across. So I'm not expecting readers to take everything in this guide at face value either (and many statements have nuances/exceptions, some of which I don't even realize should be explained to readers; it's hard to translate precise intuitions into words after all). Unfortunately, separating the signal from the noise is hard. If you're skeptical about a principle or how strongly it holds, try running it through KataHex yourself (if you have it installed), and try variations to understand the nuances, when the principle holds, and importantly ''why'' it holds. Sometimes people are skeptical of KataHex itself or think they can do better; I think there are good heuristic reasons why this thinking is misguided (the most simple one is that KataHex is far stronger than the strongest humans).

==General==

===Common mistake: playing near your own edge===

Playing near your own edge with no nearby stones is a bad idea in the opening and even middlegame. This is one of the most common mistakes among human players. You need a strong (often tactical) reason to do otherwise.
* When is playing near your own edge a good idea? There are a few cases, like when you're playing adjacent to existing stones, responding to a joseki, or pushing a ladder as the attacker, but all of these involve stones in the vicinity.
* This holds most strongly for 13×13 to 19×19, but applies more often than not to 11×11 too.
* Playing near your own edge in the opening or early middlegame should be a "last resort". I would recommend never doing so in the first 30-40 moves in 19×19 unless you have a very good reason.

===Common mistake: bad minimaxing===

Many intermediate and strong players are too eager to minimax. They play bad minimaxing moves (that KataHex thinks are inferior to directly connecting) more often than they fail to minimax. The issue is that minimaxing isn't always free; your opponent can often intrude for useful territory, and sometimes the territory your opponent gains is more useful than what you gain by minimaxing. It's a difficult skill to judge which player gains more from a minimax, but there are common patterns.
* Players frequently attempt to minimax by playing adjacent to their existing stone on the 5th or 6th row of their own edge. Playing two adjacent stones like this is not very efficient and typically a mistake.
* Blue's minimaxing attempt is unsuccessful; KataHex already thinks Red is 98% to win: https://hexworld.org/board/#15nc1,c2e11c13e10b12e9c10d8b9c7a8b6c6b7d7c8e8d9
* A common example; Red is 98% to win: https://hexworld.org/board/#13nc1,c2d10c11c10b11c9b8a10. Note that Red 7 at b8 is stronger than at a10, because b8 combines well with c2 and b5 on 13×13. On 15×15, KataHex prefers 7. a12 (the equivalent of a10 on 13×13) instead.
* A case where minimaxing is in fact the best move: https://hexworld.org/board/#13nc1,c2d10c9d11

===Common mistake: bad bridge peep===

A common scenario is the [[Peep#Bridge_peep | bridge peep]]. Blue should play at (*) as long as she thinks it's forcing enough for Red to defend his bridge.

However, even if Blue hasn't played (*) yet, Red should usually mentally place a blue stone there. Especially on larger boards, Blue will be able to play at (*) sometime in the future. This is important: even if (*) is empty, I like to think of this pattern as ''implying'' a blue stone at (*) (in the absence of tactics that somehow prevent Blue from playing there). Another way (useful for me, but the analogy might be nonsense to others) to think about it is that Blue's stone at A is well-placed enough relative to Red's bridge that she has some "potential energy" at (*).

<hexboard size="3x3"
coords="hide"
edges="hide"
contents="R c1 b3 B A:a2 E *:c2"
/>

However, 80% of the time, if there's no blue stone at A, Blue should not intrude on either side of the bridge. Intruding is usually a mistake because it settles the [[question]] for Red. If Red knows he will have a stone at A or B because Blue already intruded in the other spot, Red can better plan his future moves because of that! There are exceptions; if Blue is confident that only one of the intrusions will ever be useful for her, then intruding is often strong.

<hexboard size="3x2"
coords="hide"
edges="hide"
contents="R b1 a3 E A:a2 B:b2"
/>

===Ladders===
* Don't worry about switchbacks, climbing, or ladder creation templates on 15x15 or larger. They can be theoretically interesting and make for good puzzles, but games are rarely close enough for these to matter. I personally don't think about them on any board size, but others may reasonably disagree with this approach.
* Q: When should the defender of a ladder yield? A: Usually, don't yield unless your opponent can cut through two of your stones, one above and one below the ladder. In that case, consider yielding.
** [https://hexworld.org/board/#13nc1,c2e9c10d8b9c11d10d12 Here] is an example. If Blue instead plays 8. d11, Red 9. e10 cuts through e9 and d11.
** In [https://playhex.org/games/66b8a125-bf02-42d2-8900-af3042a8ccf8 this game] from the [[Tournaments#Hex_Monthly | Hex Monthly 21]] ([https://hexworld.org/board/#14c1,c2d11k10k4b5f10g9i9i8g10h9h10j8j9b12c10l8k8l7k7l6k11j11h12m5k6l4m4l5l9k9h13i13l3k5i5j5j4i6h6f8h7i7f9g8f7e8e7d8d7b8c8b9c9a11b13d10c11c12,c13d12d13e11:rw HexWorld link]), the yielding move 60. c14 is winning, while the played 60. c13 (which allows Red d12, cutting through Blue's c13 and d11 stones) is losing.

===Other===

* Be aggressive; always be ready to capitalize on an opponent's mistake. Here are a few illustrative examples of this and some of the above concepts:
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2368410&nmove=8 this game], 6. c9 was a mistake (and an example of incorrect minimaxing); 7. a10 is the [[Openings_on_19_x_19#4-4_obtuse_corner | standard reply]]. White should have played 8. b8; j4 was a mistake. Black's 9. b8 is the "capitalizing" move; KataHex already says 99% win rate for Black. In this case, there were several good options on move 9. These alternatives are threatening enough that Black didn't have to play b8, but this won't always be true, like in the next example.
** [https://littlegolem.net/jsp/game/game.jsp?gid=2362913&nmove=26 Here] is another example (interestingly, White resigned in a winning position according to the bot, though the final position is highly tactical). KataHex actually thinks Black has only an 35% win rate after 26. m6. However, 27. d11 is a "minimaxing with adjacent stones" mistake, bringing Black's win rate down to 3%.
** [https://www.littlegolem.net/jsp/game/game.jsp?gid=2400947&nmove=6 Here] is an example where White's win rate increased consistently from 65% after move 6 to 98% after move 16. The aggressive bridging sequence in moves 8–16 is very strong because it basically neutralizes Black's stones on d16 and d17; it is a big mistake for Black to ignore the threat here. Move 11 is also an example of a bad bridge peep.

* Don't play a move that makes your opponent's existing stones unnecessarily well-placed relative to your new stone. This is covered briefly in a couple places in the [[Openings_on_19_x_19#General_principles | 19×19 opening guide]]. Here are a couple more examples.
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2369954&nmove=26 this game] ([https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6 HexWorld link]), what should Black play on move 27? White is threatening to connect 26 back to the 16–24 group if Black isn't careful. Black's obvious options are to play at c8 and c9. Which option is better?
** A tempting answer is c8 because it appears to connect more closely to the top. However, note that White's stone on move 14 functions exactly like an opening a3 stone. For example, it can escape 2nd but not 3rd row ladders. If Black plays c8, White can play the [[A3 escape trick]], as shown in [https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6c8e7d8e8d9e9 this continuation]. This sequence allows White to "make the most" of stone 14, and Black shouldn't allow it. Better for Black is 27. c9, which doesn't allow White to carry out a 3rd-to-5th row switchback. Indeed, KataHex evaluates c9 as 97% to win for Black and c8 as only 30%.
** Consider [https://hexworld.org/board/#13n,c2d10c11c10b11e6b10d8l4l3c8d5b5d6h7h6j5i6k7 this position] (which is based on [https://www.littlegolem.net/jsp/game/game.jsp?gid=2369782 this game]). Where should White play in the lower-right acute corner? Black had just played at the "3-7" point from White's perspective; as mentioned in the 19×19 opening guide, a strong response to 3-7 is 4-4 (at j10) instead of the usual 5-4 (at j9). A player who is short on time can play j10 without thinking too hard. As it turns out, KataHex thinks j10 is the best move, evaluating it as 91% for White, whereas j9 is only around 50%.

* Sometimes, you know you have a completely winning position after your opponent blunders very early in the game. Your goal is to preserve that win until the end of the game. Don't overextend yourself and try to win too "quickly", possibly making possibly suboptimal moves in the process. Even when KataHex plays itself with a significant handicap (like playing without swap), a significant fraction of the board seems necessary to carry out the win, assuming strong defense. I prefer taking my time to "fill the board" by focusing on playing moves at least as good as my opponent's, instead of worrying about connecting in as few moves as possible.

==Local patterns==

An especially effective way to improve at Hex is to learn and gain intuition about lots of small local patterns. These patterns are small enough that they occur basically everywhere when you play Hex. Knowing the patterns can help you avoid blunders (for example, avoiding adjacent or bridge ''allow-cuts'', as explained below) and make good moves when it's otherwise unclear what to play.

===Cuts===
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 c1 R b1 1:b2"
/>
<hexboard size="1x3"
float="inline"
edges="none"
coords="none"
contents="B a1 c1 R 1:b1"
/>
<hexboard size="3x4"
float="inline"
edges="none"
coords="none"
visible="-(a1 b1 d1 a3 c3 d3)"
contents="B a2 d2 R c1 1:b3 S b2 c2 E *:(b2 c2)"
/>
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a3 c1)"
contents="B a1 c3 R 1:b2 S b1 a2 c2 b3"
/>
Consider the four patterns above. Let's call the patterns '''adjacent cut''', '''near cut''', '''short bridge cut''' (or just '''bridge cut'''), and '''long bridge cut''' respectively. In some sense, Red is "cutting" through two of Blue's stones in each pattern, so Red is the '''attacker''' and Blue is the '''defender'''. We can also use '''cut''' as an attacking verb and '''allow-cut''' as a defending verb. For example, the adjacent cut pattern is favorable for the attacker (Red above), so we can say that it's a good idea to ''adjacent-cut'' through opponent stones, and a bad idea to ''adjacent allow-cut'' (meaning it's a bad idea to play a move that allows your opponent to adjacent-cut through two of your stones).

Why is it strong to adjacent-cut (and equivalently, why is it a mistake to adjacent allow-cut)? The most intuitive reason is that if Blue allow-cuts, it takes 3 extra moves for her to connect her two stones, as shown below. In a sense, this makes it unlikely for both stones to be concurrently useful in the final connection — if one of the stones proves useful, chances are the other one won't be.
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c3)"
contents="B a2 c1 1:a3 2:b3 3:c2 R b1 b2"
/>

The (short) bridge cut is strong for the attacker (Red above) because he captures the two hexes marked (*). If Blue allow-cuts, often the damage has already been done even if Red doesn't actually cut through immediately on the next turn. (However, note that this isn't always true; some tactical situations require Red to cut through immediately instead of playing elsewhere.)
* There are situations where Blue ''should'' allow-cut. If Red's cutting move comes with concessions, like the move is on his second or third row (and especially if Red already has a stone near his own edge, so cutting through would be overplaying that edge), the pattern is much less favorable to Red.

The near cut and long bridge cut are also favorable for the attacker, but much less clearly so — there are many more "exceptions" where cutting is bad and allow-cutting is fine. On the other hand, the adjacent and short bridge cuts are quite strong, and cutting through is usually the best move (unless doing so involves playing a stone that's too close to your own edge, or another major concession).

Note that allow-cuts can occur not only ''proactively'' before the attacker plays the cutting move, but also ''retroactively'' after the "cutting" move is played (of course, it's not really a "cutting" move until after the defender allow-cuts). In other words, Blue 1 is considered an allow-cut in both of these examples (proactive on the left and retroactive on the right), and they are equally bad for Blue assuming that Red intends to cut in the proactive case:
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 1:c1 R b1"
/>
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 1:c1 R b1 b2"
/>

An important way of pruning moves and avoiding blunders is making sure you don't play moves that adjacent or bridge allow-cut, either proactively or retroactively, unless you have a good reason to do so. In particular, the retroactive case can be easy to overlook if you're not used to thinking in this way.

===Small patterns===
<hexboard size="2x2"
float="inline"
edges="none"
coords="none"
visible="-a1"
contents="B b2 R b1 1:a2"
/>
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 a3 c2 c3)"
contents="B b3 R c1 1:a2"
/>
<hexboard size="4x2"
float="inline"
edges="none"
coords="none"
visible="-(a1 b4)"
contents="B a4 R 1:b1"
/>

In each of these patterns, we will discuss the strength of Red's move 1 ''in a vacuum'', meaning in the absence of nearby stones, like near the middle of an empty board on 19×19 or even larger boards. It is most useful to analyze each pattern without the influence of nearby stones or edges (which can be thought of as virtual rows of stones).
* Pattern 1: It's typically an inaccuracy for Blue to play adjacent to a red stone in a vacuum. Red's best response is usually playing adjacent to both stones, as in Red 1.
* Pattern 2: Blue plays a bridge away from Red's stone. This is often a good move, and Red 1 a bridge away from both stones is also good.
* Pattern 3: Red 1 is a classic block. This is often fine for Red, and Blue doesn't have an obvious best local response. Blue could tenuki here.

===Capture patterns===

Know the most common [[captured cell]] patterns. It really reduces the number of moves you have to consider.

The bridge cut is one of the most important capture patterns.

* A typical example can be found in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2233736&nmove=10 this game]. After 10. g11, Black would have the easiest time with 11. h9, with 95% win rate, instead of the played k8 with 74% win rate. After k8 was played, one of White's best moves would've been h9 to prevent Black from playing the same. KataHex does think Black has other reasonable options on move 11, but they mostly appear to just delay h9.

Here is another common capture pattern. The other patterns on [[captured cell]] are useless because they have too many Red stones adjacent to each other, which is too inefficient to occur in normal play.

<hexboard size="3x3"
coords="hide"
edges="hide"
contents="R c1 c3 b3 B a2 S b2 c2 E *:(b2 c2)"
/>

This ''one-sided'' capture pattern is very common. I like to mentally replace A with a blue stone and (*) with a red stone (even though Blue won't always achieve this), because Blue should never play at (*). [[Edge template IV3a]] is useful to know here.
<hexboard size="3x2"
coords="hide"
edges="hide"
visible="-a1"
contents="R b1 a3 b3 S a2 b2 E *:b2 A:a2"
/>

The trapezoid and crescent also one-sided-capture the hexes at (*). Anecdotally, this fact is more useful on smaller boards like 11×11, but it's less common overall than the previous capture patterns.
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
visible="-a1"
contents="R b1 a3 c3 c1 S a2--c2 b3 E *:(c2 b3)"
/>
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
visible="-a1"
contents="R a2 b1 c1 c3 S b2 c2 a3 b3 E *:c2"
/>

==How to think about the opening and middlegame==

Some of these points sound like hyperbole, but they are a fairly accurate representation of how I think about Hex, especially on larger boards. There are exceptions, but exceptions are rarer than one might think.

* In the first half of the game, focus all of your attention into playing "efficient" moves. These are moves that work well with existing stones and board edges. Excellent intuition of efficient patterns is enough to get strong middlegame positions against even top players (as of 2024).
* Don't worry about tactics until the late middlegame. It's a waste of time to analyze sequences >10 moves deep. In the opening, the best move you can find with only intuition is no worse than the best move you can find with calculation.
** Why? Compared to bots, humans are very bad at calculation. Your intuition (which you learn directly from KataHex) is more useful than hypothetical hours or days of error-prone human analysis.
* Use your knowledge of efficient patterns to plan ahead. This is where you want to do shallow analysis, typically 1-5 moves deep. A good move does as much as possible:
** allows you to play efficient moves later
** prevents your opponent from playing an efficient move outright
** allows your opponent an efficient move, but only with concessions (say, a locally efficient but globally inefficient move, or a move that's efficient with relative to one stone but inefficient relative to another)

These points require a lot of elaboration. What do I mean by "efficient"? (Not to be confused with [[efficiency]].) There are two broad notions, '''global''' and '''local''' efficiency.
* Globally efficient stone: Broadly, any move that isn't too close to your own edge. Moves that are strong in the absence of nearby stones (like on an empty board). Corner moves are very efficient. On 19×19, moves near your opponent's 5th row are very efficient.
* Locally efficient stone: In the presence of nearby stones, some configurations are broadly "good" or "bad" for one player. They aren't ''always'' good or bad because of tactics, but because humans can't calculate everything, it's extremely useful to have decent heuristics that work most of the time.

In a sense, you want to play the move that maximizes the "sum" of local and global efficiency. This isn't an exact science; use your intuition to make an educated guess.
* The best move locally is rarely the best move globally. Hex is a game of concessions and tradeoffs. There is no "free lunch," unless you count concepts like inferiority/domination.
* Conversely, if it looks like your opponent is beating you on one side of the board, there are two possibilities:
*# You played suboptimal moves in that region.
*# Your opponent overplayed that region, which necessarily means they made concessions elsewhere (usually the opposite side of the board). This means your opponent ''underplayed'' the other region.
* If option 1 is true, you are probably losing anyways, so assume (or hope) option 2 is true. If you opponent overplays region A and underplays region B, it's typically a good idea to focus your play in region B. Why? If you keep playing in region A, you're potentially helping your opponent out of their mistake, by making your opponent less over-connected in region A. For example, your opponent could have a minimaxing move that makes them much stronger in region B, at the "concession" of making them weaker in region A — but this isn't really a concession because your opponent had strength to spare in region A.

The discussion so far has been highly abstract. What does efficient play look like in practice? As explained in the "Cuts" section, it's strong to "cut through" your opponent's potential connection, especially in the adjacent cut and bridge cut.

The small patterns in the "Capture patterns" section above are also locally efficient for the capturing player. You should generally play them when given the opportunity, and prevent your opponent from doing the same.

Just as offense and defense are equivalent in Hex, so are playing locally efficient moves and inducing your opponent to play locally ''inefficient'' moves. Sometimes, locally inefficient patterns arise when one player induces the other to play a bunch of stones close together. Here is an example (which occasionally happens in actual games), where Red is connected but took a lot of stones to connect.
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
contents="R b1 c1 a3 B a2 R 1:b3 B 2:c2 R 3:b2"
/>

In the following examples, P1 and P2 are efficient patterns for Red relative the corresponding edges, while P3 is inefficient. P1 is common when Red opens with a first column opening. P2 requires elaboration: normally Red 1 is a weak move because it's too close to Red's own edge. However, when Red must play on his first 3 rows (because of very limited space), Red 1 or its mirror image at (*) is usually the best, because it prevents Blue from playing there and forming the equivalent of P1 herself. P2 also occurs in theoretical contexts, like [[Template Va#If Blue moves at y:]]. In P3, Red 1 is a tempting (because of [[edge template III1b]]), common, but inefficient move, because Blue's response at one of (*) is usually strong. If Red must play near his own edge, he should instead play as in P2 above.
P1: <hexboard size="3x2"
float="inline"
edges="left"
coords="none"
contents="R a2 1:b3"
/>
P2: <hexboard size="2x6"
float="inline"
edges="bottom"
coords="none"
contents="B c2 R 1:e1 E *:b1"
/>
P3: <hexboard size="3x5"
float="inline"
edges="bottom"
coords="none"
contents="B c3 R 1:d1 E *:(b2 e2)"
/>

===Making inferences: An extended example===

Use the patterns and intuitions that you know, to reason and make inferences in unfamiliar positions. You might not always find the best move, but you can often prune bad moves. Here is an extended example, involving several closely related patterns that are useful to know in isolation.
# ''Pattern:'' If you play the 4-4 3-3 obtuse corner joseki, it is important to know that Blue 6 [https://hexworld.org/board/#19nc1,d16c17d17c18d18e14c15 here] is weak.
#* This is because Red 7 undermines Blue 6 and also reduces Blue 2/4 from a 3rd row ladder escape to a 2nd row escape. Also, Blue's potential intrusion at c16 doesn't gain her anything.
#* Instead, [https://hexworld.org/board/#19nc1,d16c17d17c18d18d15 here] is a good local response for Blue. This gives Blue a 5th row ladder escape while blocking Red 1.
#* On the other hand, if Blue only has a 2nd row escape and doesn't have the "potential energy" to get a 3rd row escape, you might guess that Blue 6 above becomes a good local move, because Red 7 loses one of its main strengths. Such a guess would be correct: in [https://hexworld.org/board/#19nc1,d16c17p15d17e16g16e17f18d18e14c15 this position], Blue 10 is a strong move. On the other hand, Blue 10 at d15 would be weak here, as it's quite wasteful, only converting a 2nd row escape into a 3rd row escape.
# ''Pattern:'' If Red tenukis on move 3, then Blue 4 is locally strong in [https://hexworld.org/board/#19nc1,d16e14:pf15 this position]. In particular, Blue 4 is much stronger than bridging directly towards Blue's own edge with g13, because the former is a strong blocking move that reduces the effectiveness of Red 1. This strength is in spite of Red's intrusion on move 5. (I first saw this move in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2090091&nmove=16 this bot game] from [https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=739 this thread on LittleGolem].)
#* Variations of this pattern appear frequently in obtuse corner joseki. If Red started with the 5-5 obtuse corner, then the equivalent Blue 4 is [https://hexworld.org/board/#19nc1,e15f13:pg14 also] strong.
#* Another simple example is Red 5 in [https://hexworld.org/board/#19nc1,d16d15f14:pe13 this position].
# ''Inference:'' Let us consider [https://hexworld.org/board/#19nc1,d16c17d17c18d18:p this position] again. Suppose Blue tenukis on move 6. What is a good local response for Red on move 7? There is more than one right answer, but you just need to find one move that isn't a blunder.
#* ''Hint:'' Use the two patterns above to come up with an answer.
#* ''Hint:'' Consider that Blue has a 3rd row ladder escape, so Blue d15 is locally strong but Blue e14 is weak. This is the case for future blue stones in this local region, even though Blue already played elsewhere on move 6. Therefore, Red should perhaps try to play a move that makes Blue d15 less attractive, even if it comes at the concession of making Blue e14 more attractive (since the latter is ''a priori'' a weak move, such a concession gives up less than it otherwise might).
#* ''Answer:'' Red 7 [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13 here] is a good local move.
#** Why? Ignoring stones 2-5, the presence of Red 1 and 7 makes Blue e14 more attractive. Red's 4-4 stone, much like the obtuse corner opening, tends to shift Blue's efficient stone locations up one row. With that in mind, Red's move 7 is a bit like Red e14 on an empty board, where Blue e15 is [https://hexworld.org/board/#19nc1,e14e15 the best response]. However, stones 2-5 make Blue e14 a bad move, despite Red 7 making it slightly less bad (this is the concession-but-not-really-a-concession that Red intended).
#** Let's now consider Blue d15. Normally a decent local move after stones 1-5, it becomes quite weak after Red 7, because of [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13d15f14 this sequence]. This is a typical example of the second pattern in this extended example. (Actually, if Blue had two moves in a row in this region, Blue 8 wouldn't be so bad because she could follow up with f14, where Red wants to play move 9. This is a tactical situation where it's bad for Red to tenuki.)
#** If we wanted to be complete, Blue actually has another option for move 8, f14 first which threatens d15: [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13f14d15d14e14 here]. This is an example of the "skew cut" from above. Red should respond with d15 himself, or else Blue can play d15. This is probably Blue's best local response.
#** Every move makes a concession, but some concessions are more effective than others. As alluded to in the hint, Red 7 weakens a strong Blue reply (d15) while strengthening a weak Blue reply (e14). Red's goal is to minimize the strength of Blue's ''best'' response (rather than the strength of Blue's average response), and Red 7 at e13 accomplishes that beautifully.

Part of getting stronger at Hex (as with other games) is being able to compress knowledge and patterns effectively. If you understand this extended example, you can infer the best move in many similar but novel situations, without memorizing each one individually. If you forget which move is exactly the best, you can recover the right move with high probability by just reasoning about related patterns that you do know are strong or weak. The "A3 escape trick" example earlier in this article is another instance of this concept.

On larger boards and particularly near the opening or middlegame, when you have a lot of time to think about a position, it may be worth spending a good chunk of it making inferences and scoping out moves that are either good or to be avoided by each side, rather than spending most of your time calculating tons of lines and variations.
* In fact, once you've scoped out that "moves A and B are good for Red, while moves X and Y are bad for Red" and similarly for Blue, that makes the process of analyzing and pruning lines much more efficient.

==Joseki==

* The 4-3 corner move is almost always a mistake compared to the 5-4, even on 11×11. If your opponent plays 4-3, 90% of the time the best local reply is 3-3. If 4-3 is played, here are some common sequences:
<hexboard size="5x7"
coords="hide"
edges="bottom right"
contents="R 1:e2 B 2:e3 R 3:d3"
/>

<hexboard size="5x7"
coords="hide"
edges="bottom right"
contents="R 1:e2 B 2:e3 R 3:f2 B 4:f3 R 5:d3 E a:b4 b:c5"
/>

In the second joseki, Blue should play either (a) or (b), or she can defer the question until later. On larger boards with no nearby stones, (b) is often better.

* If your opponent plays the 5-4 joseki, a reasonable choice is to always play the 4-4 "high intrusion." KataHex prefers the 4-4 response by far, especially on 13×13 and larger. It's easily the "safest" choice, and many strong players have recently shown a clear preference for 4-4.

* In [https://hexworld.org/board/#14nc1,k10k11l10l12m11m12j11i13h12g14f13g12g13 this often-played joseki], Red 7 is a mistake because of the territory Blue gets after move 8.
** If the obtuse corner on Red's same edge is empty, a more principled response for Red is to play in that corner, like [https://hexworld.org/board/#14c1,k10k11l10l12d11 this].
** If the obtuse corner is already occupied by Blue, then Red should still respond in the obtuse corner. One reasonable option (among others) is [https://hexworld.org/board/#14c1,:pd11k10k11l10l12e11 this].

* See more discussion of acute corner joseki at [[Openings_on_19_x_19#Acute_corner_theory]]. For the obtuse corner, see [[Openings_on_19_x_19#Obtuse_corner_theory]] which covers responses to the 4-4 and 5-5 obtuse corner in great detail. Most of this is applicable to 13×13 and larger, and not just 19×19.

==Specific tactics==
===Threatening a bridge cut===
One of my favorite tactics is threatening a bridge cut, where the opponent needs to play too close to their own edge to defend:
<hexboard size="8x8"
float="inline"
edges="bottom left"
coords="hide"
contents="B *:d5 g2 R 1:f4 B 2:e3"
/>
Red 1 is a strong move, because it forces Blue to respond at 2 (or else Red will play the bridge cut there), but Blue 2 is normally overplaying on the left side when combined with Blue's stone at (*). Be on the lookout for this when your opponent plays the 4-4 and 7-7 obtuse corner, or 5-5 and 8-8.

This tactic can work well in other configurations, like [https://hexworld.org/board/#19c1,:pg8:pg11h9 this] and [https://hexworld.org/board/#19c1,:pe9:ph9g8 this]. However, be careful when using this tactic! When Blue isn't actually making a concession by bridging her two stones, like when the pattern occurs further from her own edge, it can be a blunder for Red to threaten a bridge cut. [https://hexworld.org/board/#19nc1,:pg8:pj5i7h6 Here] is an example where Red 5 is a blunder in the absence of nearby stones.

[https://www.littlegolem.net/jsp/game/game.jsp?gid=2436000&nmove=12 Here] is an example from a real game. Black's best move is 13. h5 (in the actual game, Black doesn't play this until move 19 but is still arguably rewarded for doing so). One might think that h5 is too strong near the top edge because of c2 and b5, but this is not the case because the white stone on c5 partially neutralizes b5.

[[category:strategy]]

Strategic advice from KataHex

2025-11-27T02:18:39Z

Hexanna: elaborate on allow-cuts

==Introduction==

Several hex bots have popped up in the last few years. Many are neural nets and adaptations of Go bots like AlphaGo Zero. The most recent (and likely strongest) as of 2025 is [[KataHex]], which is based on the open-source KataGo. These bots have influenced human play in many aspects already; one of the most notable is the bots' preference for the 4-4 obtuse corner over the 5-5. Many old strategy guides have become partially (if not mostly) outdated.

This article attempts to bring some strategy advice "up-to-date". There's a lot of random insights; most are a direct result of playing around with KataHex evaluations for thousands of hours, including analyzing the games of strong players for common mistakes. Since this is strategic advice and not tactics, the advice won't apply 100% of the time, and many players may disagree with some parts. Win percentages and evaluations in this article are based on KataHex net [https://github.com/hzyhhzy/KataGo/releases/tag/Hex_20240812 20240812], which is significantly stronger than previous nets.

Care has been taken to only draw conclusions when there are multiple examples of KataHex preferring something. However, the main goal is to give actionable advice to make players stronger, and there isn't rigorous proof for basically any of these statements.

===Big picture, and some meta-points===

* '''TL;DR:''' What are the most important principles? Even if you learn nothing else, you should learn these:
*# Avoid playing close to your own edge, especially in the opening, unless you have a very good reason to do so.
*#* Think of your own edge as a whole row of stones of your color. You already have a bunch of stones (and thus strength) there, and playing yet another one close by would be a waste. That stone is much better served elsewhere, where you don't have as much strength.
*# Bridges are the most efficient way to connect your own stones—not adjacent stones, trapezoids, crescents, or hammocks. If there are no opposing stones nearby, playing adjacent to an existing stone of your color is usually a mistake; if you want to do that, consider whether there's a better move a bridge away from that stone.
*#* If there are nearby opposing stones, things become more nuanced, but bridging to your existing stones is often still a good idea.
*# Usually, cut through an opponent's bridge if given the opportunity, and avoid letting it happen to you. In the pattern below, Red cuts through Blue's two stones a bridge apart. This is very good for Red and bad for Blue, because it's very hard for Blue to connect the two stones now.
*#* <hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 c1 R b1 1:b2"
/>
* This guide assumes you've read [http://www.mseymour.ca/hex_book/hexstrat.html ''Hex: A Strategy Guide''] or have equivalent knowledge.
** ''Hex: A Strategy Guide'' predates KataHex, but it's very well made, and 90% of the advice is still good.
* On smaller boards, tactics and raw calculation matter more. On larger boards, strategy matters more. (It's ultimately a spectrum, and it varies by skill level: for beginners, size 13 might be considered large; for strong players, tactics can matter a great deal on size 13.)
** On smaller boards, the exact board size affects strategy (some moves can be good on 11×11 but not 12×12). The larger the board, the less the exact board size matters. The general strategy for 19×19 and 20×20 should be nearly identical.
** On smaller boards, there is more space close to corners and edges. On larger boards, there is more "open" space (hexes not close to corners and edges). I intentionally said "open" space and not "center" space, because there is not much special about hexes near the center of the board, except that they are far away from corners and edges. But on large boards, there are many such areas.
** In particular, the larger the board size (starting at around size 13–15), the more important it is to know how to play in "open" areas. Many existing guides cover corner/edge strategy heavily—and correctly so, as corners/edges remain important up to very large sizes (25–30 or so). People read those guides and often know how to play the first 10–20 moves of a 17×17 game, but often not after that, when most of the game is still remaining!
** Strategy can also be ''local'' or ''global''. Local strategy tells you how to play within a small region of the board, and global strategy tells you how to play in relation to the whole board (or across multiple local regions). Both are important.
** The strategy of "open" areas is mostly local, and so the larger the board, the higher the relative importance of local strategy. Why?
*** On a large Hex board, the "middle" is relatively homogeneous—two separate open areas near the middle are basically the same in terms of how you should play them; it doesn't matter that one of the areas is 15 hexes away from an edge and the other one 18. Because of this homogeneity, the same basic principles will apply multiple times in the span of a game.
*** Knowing how to play across multiple local regions is very hard, and it's often different across games (especially for humans). It's better to master local play first, since similar local play principles will be relevant every game.
* How should a game of Hex (on larger boards) broadly feel like?
** While the win condition is binary (you either connect or you don't), the actual gameplay feels like a gradual accumulation of tiny advantages and disadvantages. Strong players often "know" they're winning in the middlegame, not because they see an actual way to connect, but because their opponent made some mistakes, however small, along the way. (You rarely know for sure who's winning until the end, and you're ultimately making an educated guess.)
** In particular, it's usually very hard to tell who's winning just by looking at a position. Contrast that with games like chess, where you can often tell by counting up material and broadly looking at positional factors. In Hex, it's much tractable to tell how good your position is ''relative to a few moves ago.''
** Relatedly, there's (almost) no meaningful notion of "long-term strategy" in Hex. If anything, it can be harmful to create a long-term plan and stubbornly stick to it, because the moves you should play often differ greatly depending on your opponent's responses.
** Hex strategy is equally what to play and what ''not'' to play. The latter is frequently underemphasized. Each stone is worth a lot, and even one bad move can "undo" 10–20 good ones. The risk-reward balance strongly favors avoiding mistakes, rather than attempting risky but potentially "brilliant" moves.
* Now some points about how to learn. From what I observe, people often read strategy advice and then fail to follow it consistently. This is often either because of skepticism, or not realizing just how often particular pieces of advice should be followed (it's tempting to think that you've found the exception to the rule).
** It's healthy to be skeptical—there is a lot of ''bad'' strategy advice, and it would probably do more harm than good to implement every piece of advice you come across. So I'm not expecting readers to take everything in this guide at face value either (and many statements have nuances/exceptions, some of which I don't even realize should be explained to readers; it's hard to translate precise intuitions into words after all). Unfortunately, separating the signal from the noise is hard. If you're skeptical about a principle or how strongly it holds, try running it through KataHex yourself (if you have it installed), and try variations to understand the nuances, when the principle holds, and importantly ''why'' it holds. Sometimes people are skeptical of KataHex itself or think they can do better; I think there are good heuristic reasons why this thinking is misguided (the most simple one is that KataHex is far stronger than the strongest humans).

==General==

===Common mistake: playing near your own edge===

Playing near your own edge with no nearby stones is a bad idea in the opening and even middlegame. This is one of the most common mistakes among human players. You need a strong (often tactical) reason to do otherwise.
* When is playing near your own edge a good idea? There are a few cases, like when you're playing adjacent to existing stones, responding to a joseki, or pushing a ladder as the attacker, but all of these involve stones in the vicinity.
* This holds most strongly for 13×13 to 19×19, but applies more often than not to 11×11 too.
* Playing near your own edge in the opening or early middlegame should be a "last resort". I would recommend never doing so in the first 30-40 moves in 19×19 unless you have a very good reason.

===Common mistake: bad minimaxing===

Many intermediate and strong players are too eager to minimax. They play bad minimaxing moves (that KataHex thinks are inferior to directly connecting) more often than they fail to minimax. The issue is that minimaxing isn't always free; your opponent can often intrude for useful territory, and sometimes the territory your opponent gains is more useful than what you gain by minimaxing. It's a difficult skill to judge which player gains more from a minimax, but there are common patterns.
* Players frequently attempt to minimax by playing adjacent to their existing stone on the 5th or 6th row of their own edge. Playing two adjacent stones like this is not very efficient and typically a mistake.
* Blue's minimaxing attempt is unsuccessful; KataHex already thinks Red is 98% to win: https://hexworld.org/board/#15nc1,c2e11c13e10b12e9c10d8b9c7a8b6c6b7d7c8e8d9
* A common example; Red is 98% to win: https://hexworld.org/board/#13nc1,c2d10c11c10b11c9b8a10. Note that Red 7 at b8 is stronger than at a10, because b8 combines well with c2 and b5 on 13×13. On 15×15, KataHex prefers 7. a12 (the equivalent of a10 on 13×13) instead.
* A case where minimaxing is in fact the best move: https://hexworld.org/board/#13nc1,c2d10c9d11

===Common mistake: bad bridge peep===

A common scenario is the [[Peep#Bridge_peep | bridge peep]]. Blue should play at (*) as long as she thinks it's forcing enough for Red to defend his bridge.

However, even if Blue hasn't played (*) yet, Red should usually mentally place a blue stone there. Especially on larger boards, Blue will be able to play at (*) sometime in the future. This is important: even if (*) is empty, I like to think of this pattern as ''implying'' a blue stone at (*) (in the absence of tactics that somehow prevent Blue from playing there). Another way (useful for me, but the analogy might be nonsense to others) to think about it is that Blue's stone at A is well-placed enough relative to Red's bridge that she has some "potential energy" at (*).

<hexboard size="3x3"
coords="hide"
edges="hide"
contents="R c1 b3 B A:a2 E *:c2"
/>

However, 80% of the time, if there's no blue stone at A, Blue should not intrude on either side of the bridge. Intruding is usually a mistake because it settles the [[question]] for Red. If Red knows he will have a stone at A or B because Blue already intruded in the other spot, Red can better plan his future moves because of that! There are exceptions; if Blue is confident that only one of the intrusions will ever be useful for her, then intruding is often strong.

<hexboard size="3x2"
coords="hide"
edges="hide"
contents="R b1 a3 E A:a2 B:b2"
/>

===Ladders===
* Don't worry about switchbacks, climbing, or ladder creation templates on 15x15 or larger. They can be theoretically interesting and make for good puzzles, but games are rarely close enough for these to matter. I personally don't think about them on any board size, but others may reasonably disagree with this approach.
* Q: When should the defender of a ladder yield? A: Usually, don't yield unless your opponent can cut through two of your stones, one above and one below the ladder. In that case, consider yielding.
** [https://hexworld.org/board/#13nc1,c2e9c10d8b9c11d10d12 Here] is an example. If Blue instead plays 8. d11, Red 9. e10 cuts through e9 and d11.
** In [https://playhex.org/games/66b8a125-bf02-42d2-8900-af3042a8ccf8 this game] from the [[Tournaments#Hex_Monthly | Hex Monthly 21]] ([https://hexworld.org/board/#14c1,c2d11k10k4b5f10g9i9i8g10h9h10j8j9b12c10l8k8l7k7l6k11j11h12m5k6l4m4l5l9k9h13i13l3k5i5j5j4i6h6f8h7i7f9g8f7e8e7d8d7b8c8b9c9a11b13d10c11c12,c13d12d13e11:rw HexWorld link]), the yielding move 60. c14 is winning, while the played 60. c13 (which allows Red d12, cutting through Blue's c13 and d11 stones) is losing.

===Other===

* Be aggressive; always be ready to capitalize on an opponent's mistake. Here are a few illustrative examples of this and some of the above concepts:
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2368410&nmove=8 this game], 6. c9 was a mistake (and an example of incorrect minimaxing); 7. a10 is the [[Openings_on_19_x_19#4-4_obtuse_corner | standard reply]]. White should have played 8. b8; j4 was a mistake. Black's 9. b8 is the "capitalizing" move; KataHex already says 99% win rate for Black. In this case, there were several good options on move 9. These alternatives are threatening enough that Black didn't have to play b8, but this won't always be true, like in the next example.
** [https://littlegolem.net/jsp/game/game.jsp?gid=2362913&nmove=26 Here] is another example (interestingly, White resigned in a winning position according to the bot, though the final position is highly tactical). KataHex actually thinks Black has only an 35% win rate after 26. m6. However, 27. d11 is a "minimaxing with adjacent stones" mistake, bringing Black's win rate down to 3%.
** [https://www.littlegolem.net/jsp/game/game.jsp?gid=2400947&nmove=6 Here] is an example where White's win rate increased consistently from 65% after move 6 to 98% after move 16. The aggressive bridging sequence in moves 8–16 is very strong because it basically neutralizes Black's stones on d16 and d17; it is a big mistake for Black to ignore the threat here. Move 11 is also an example of a bad bridge peep.

* Don't play a move that makes your opponent's existing stones unnecessarily well-placed relative to your new stone. This is covered briefly in a couple places in the [[Openings_on_19_x_19#General_principles | 19×19 opening guide]]. Here are a couple more examples.
** In [https://littlegolem.net/jsp/game/game.jsp?gid=2369954&nmove=26 this game] ([https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6 HexWorld link]), what should Black play on move 27? White is threatening to connect 26 back to the 16–24 group if Black isn't careful. Black's obvious options are to play at c8 and c9. Which option is better?
** A tempting answer is c8 because it appears to connect more closely to the top. However, note that White's stone on move 14 functions exactly like an opening a3 stone. For example, it can escape 2nd but not 3rd row ladders. If Black plays c8, White can play the [[A3 escape trick]], as shown in [https://hexworld.org/board/#15n,a7:sd12k12l4h2g2h1c4d3c3d2c2c5d4c14b14c13b13c12b12c11b11c10b10d6c8e7d8e8d9e9 this continuation]. This sequence allows White to "make the most" of stone 14, and Black shouldn't allow it. Better for Black is 27. c9, which doesn't allow White to carry out a 3rd-to-5th row switchback. Indeed, KataHex evaluates c9 as 97% to win for Black and c8 as only 30%.
** Consider [https://hexworld.org/board/#13n,c2d10c11c10b11e6b10d8l4l3c8d5b5d6h7h6j5i6k7 this position] (which is based on [https://www.littlegolem.net/jsp/game/game.jsp?gid=2369782 this game]). Where should White play in the lower-right acute corner? Black had just played at the "3-7" point from White's perspective; as mentioned in the 19×19 opening guide, a strong response to 3-7 is 4-4 (at j10) instead of the usual 5-4 (at j9). A player who is short on time can play j10 without thinking too hard. As it turns out, KataHex thinks j10 is the best move, evaluating it as 91% for White, whereas j9 is only around 50%.

* Sometimes, you know you have a completely winning position after your opponent blunders very early in the game. Your goal is to preserve that win until the end of the game. Don't overextend yourself and try to win too "quickly", possibly making possibly suboptimal moves in the process. Even when KataHex plays itself with a significant handicap (like playing without swap), a significant fraction of the board seems necessary to carry out the win, assuming strong defense. I prefer taking my time to "fill the board" by focusing on playing moves at least as good as my opponent's, instead of worrying about connecting in as few moves as possible.

==Local patterns==

An especially effective way to improve at Hex is to learn and gain intuition about lots of small local patterns. These patterns are small enough that they occur basically everywhere when you play Hex. Knowing the patterns can help you avoid blunders (for example, avoiding adjacent or bridge ''allow-cuts'', as explained below) and make good moves when it's otherwise unclear what to play.

===Cuts===
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 c1 R b1 1:b2"
/>
<hexboard size="1x3"
float="inline"
edges="none"
coords="none"
contents="B a1 c1 R 1:b1"
/>
<hexboard size="3x4"
float="inline"
edges="none"
coords="none"
visible="-(a1 b1 d1 a3 c3 d3)"
contents="B a2 d2 R c1 1:b3 S b2 c2 E *:(b2 c2)"
/>
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a3 c1)"
contents="B a1 c3 R 1:b2 S b1 a2 c2 b3"
/>
Consider the four patterns above. Let's call the patterns '''adjacent cut''', '''near cut''', '''short bridge cut''' (or just '''bridge cut'''), and '''long bridge cut''' respectively. In some sense, Red is "cutting" through two of Blue's stones in each pattern, so Red is the '''attacker''' and Blue is the '''defender'''. We can also use '''cut''' as an attacking verb and '''allow-cut''' as a defending verb. For example, the adjacent cut pattern is favorable for the attacker (Red above), so we can say that it's a good idea to ''adjacent-cut'' through opponent stones, and a bad idea to ''adjacent allow-cut'' (meaning it's a bad idea to play a move that allows your opponent to adjacent-cut through two of your stones).

Why is it strong to adjacent-cut (and equivalently, why is it a mistake to adjacent allow-cut)? The most intuitive reason is that if Blue allow-cuts, it takes 3 extra moves for her to connect her two stones, as shown below. In a sense, this makes it unlikely for both stones to be concurrently useful in the final connection — if one of the stones proves useful, chances are the other one won't be.
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c3)"
contents="B a2 c1 1:a3 2:b3 3:c2 R b1 b2"
/>

The (short) bridge cut is strong for the attacker (Red above) because he captures the two hexes marked (*). If Blue allow-cuts, often the damage has already been done even if Red doesn't actually cut through immediately on the next turn. (However, note that this isn't always true; some tactical situations require Red to cut through immediately instead of playing elsewhere.)
* However, there are situations where Blue should allow-cut. If Red's cutting move comes with concessions, like the move is on his second or third row (and especially if Red already has a stone near his own edge, so cutting through would be overplaying that edge), the pattern is much less favorable to Red.

The near cut and long bridge cut are also favorable for the attacker, but much less clearly so — there are many more "exceptions" where cutting is bad and allow-cutting is fine. On the other hand, the adjacent and short bridge cuts are quite strong, and cutting through is usually the best move (unless doing so involves playing a stone that's too close to your own edge, or some other concession).

Note that allow-cuts can occur not only ''proactively'' before the attacker plays the cutting move, but also ''retroactively'' after the "cutting" move is played (of course, it's not really a "cutting" move until after the defender allow-cuts). In other words, Blue 1 is considered an allow-cut in both of these examples (proactive on the left and retroactive on the right), and they are equally bad for Blue assuming that Red intends to cut in the proactive case:
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 1:c1 R b1"
/>
<hexboard size="2x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 c2)"
contents="B a2 1:c1 R b1 b2"
/>

In fact, an important way of pruning moves and avoiding blunders is making sure you don't play moves that adjacent or bridge allow-cut, either proactively or retroactively, unless you have a good reason to do so. The retroactive case in particular can be easy to overlook if you're not used to thinking in this way.

===Small patterns===
<hexboard size="2x2"
float="inline"
edges="none"
coords="none"
visible="-a1"
contents="B b2 R b1 1:a2"
/>
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
visible="-(a1 a3 c2 c3)"
contents="B b3 R c1 1:a2"
/>
<hexboard size="4x2"
float="inline"
edges="none"
coords="none"
visible="-(a1 b4)"
contents="B a4 R 1:b1"
/>

In each of these patterns, we will discuss the strength of Red's move 1 ''in a vacuum'', meaning in the absence of nearby stones, like near the middle of an empty board on 19×19 or even larger boards. It is most useful to analyze each pattern without the influence of nearby stones or edges (which can be thought of as virtual rows of stones).
* Pattern 1: It's typically an inaccuracy for Blue to play adjacent to a red stone in a vacuum. Red's best response is usually playing adjacent to both stones, as in Red 1.
* Pattern 2: Blue plays a bridge away from Red's stone. This is often a good move, and Red 1 a bridge away from both stones is also good.
* Pattern 3: Red 1 is a classic block. This is often fine for Red, and Blue doesn't have an obvious best local response. Blue could tenuki here.

===Capture patterns===

Know the most common [[captured cell]] patterns. It really reduces the number of moves you have to consider.

The bridge cut is one of the most important capture patterns.

* A typical example can be found in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2233736&nmove=10 this game]. After 10. g11, Black would have the easiest time with 11. h9, with 95% win rate, instead of the played k8 with 74% win rate. After k8 was played, one of White's best moves would've been h9 to prevent Black from playing the same. KataHex does think Black has other reasonable options on move 11, but they mostly appear to just delay h9.

Here is another common capture pattern. The other patterns on [[captured cell]] are useless because they have too many Red stones adjacent to each other, which is too inefficient to occur in normal play.

<hexboard size="3x3"
coords="hide"
edges="hide"
contents="R c1 c3 b3 B a2 S b2 c2 E *:(b2 c2)"
/>

This ''one-sided'' capture pattern is very common. I like to mentally replace A with a blue stone and (*) with a red stone (even though Blue won't always achieve this), because Blue should never play at (*). [[Edge template IV3a]] is useful to know here.
<hexboard size="3x2"
coords="hide"
edges="hide"
visible="-a1"
contents="R b1 a3 b3 S a2 b2 E *:b2 A:a2"
/>

The trapezoid and crescent also one-sided-capture the hexes at (*). Anecdotally, this fact is more useful on smaller boards like 11×11, but it's less common overall than the previous capture patterns.
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
visible="-a1"
contents="R b1 a3 c3 c1 S a2--c2 b3 E *:(c2 b3)"
/>
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
visible="-a1"
contents="R a2 b1 c1 c3 S b2 c2 a3 b3 E *:c2"
/>

==How to think about the opening and middlegame==

Some of these points sound like hyperbole, but they are a fairly accurate representation of how I think about Hex, especially on larger boards. There are exceptions, but exceptions are rarer than one might think.

* In the first half of the game, focus all of your attention into playing "efficient" moves. These are moves that work well with existing stones and board edges. Excellent intuition of efficient patterns is enough to get strong middlegame positions against even top players (as of 2024).
* Don't worry about tactics until the late middlegame. It's a waste of time to analyze sequences >10 moves deep. In the opening, the best move you can find with only intuition is no worse than the best move you can find with calculation.
** Why? Compared to bots, humans are very bad at calculation. Your intuition (which you learn directly from KataHex) is more useful than hypothetical hours or days of error-prone human analysis.
* Use your knowledge of efficient patterns to plan ahead. This is where you want to do shallow analysis, typically 1-5 moves deep. A good move does as much as possible:
** allows you to play efficient moves later
** prevents your opponent from playing an efficient move outright
** allows your opponent an efficient move, but only with concessions (say, a locally efficient but globally inefficient move, or a move that's efficient with relative to one stone but inefficient relative to another)

These points require a lot of elaboration. What do I mean by "efficient"? (Not to be confused with [[efficiency]].) There are two broad notions, '''global''' and '''local''' efficiency.
* Globally efficient stone: Broadly, any move that isn't too close to your own edge. Moves that are strong in the absence of nearby stones (like on an empty board). Corner moves are very efficient. On 19×19, moves near your opponent's 5th row are very efficient.
* Locally efficient stone: In the presence of nearby stones, some configurations are broadly "good" or "bad" for one player. They aren't ''always'' good or bad because of tactics, but because humans can't calculate everything, it's extremely useful to have decent heuristics that work most of the time.

In a sense, you want to play the move that maximizes the "sum" of local and global efficiency. This isn't an exact science; use your intuition to make an educated guess.
* The best move locally is rarely the best move globally. Hex is a game of concessions and tradeoffs. There is no "free lunch," unless you count concepts like inferiority/domination.
* Conversely, if it looks like your opponent is beating you on one side of the board, there are two possibilities:
*# You played suboptimal moves in that region.
*# Your opponent overplayed that region, which necessarily means they made concessions elsewhere (usually the opposite side of the board). This means your opponent ''underplayed'' the other region.
* If option 1 is true, you are probably losing anyways, so assume (or hope) option 2 is true. If you opponent overplays region A and underplays region B, it's typically a good idea to focus your play in region B. Why? If you keep playing in region A, you're potentially helping your opponent out of their mistake, by making your opponent less over-connected in region A. For example, your opponent could have a minimaxing move that makes them much stronger in region B, at the "concession" of making them weaker in region A — but this isn't really a concession because your opponent had strength to spare in region A.

The discussion so far has been highly abstract. What does efficient play look like in practice? As explained in the "Cuts" section, it's strong to "cut through" your opponent's potential connection, especially in the adjacent cut and bridge cut.

The small patterns in the "Capture patterns" section above are also locally efficient for the capturing player. You should generally play them when given the opportunity, and prevent your opponent from doing the same.

Just as offense and defense are equivalent in Hex, so are playing locally efficient moves and inducing your opponent to play locally ''inefficient'' moves. Sometimes, locally inefficient patterns arise when one player induces the other to play a bunch of stones close together. Here is an example (which occasionally happens in actual games), where Red is connected but took a lot of stones to connect.
<hexboard size="3x3"
float="inline"
edges="none"
coords="none"
contents="R b1 c1 a3 B a2 R 1:b3 B 2:c2 R 3:b2"
/>

In the following examples, P1 and P2 are efficient patterns for Red relative the corresponding edges, while P3 is inefficient. P1 is common when Red opens with a first column opening. P2 requires elaboration: normally Red 1 is a weak move because it's too close to Red's own edge. However, when Red must play on his first 3 rows (because of very limited space), Red 1 or its mirror image at (*) is usually the best, because it prevents Blue from playing there and forming the equivalent of P1 herself. P2 also occurs in theoretical contexts, like [[Template Va#If Blue moves at y:]]. In P3, Red 1 is a tempting (because of [[edge template III1b]]), common, but inefficient move, because Blue's response at one of (*) is usually strong. If Red must play near his own edge, he should instead play as in P2 above.
P1: <hexboard size="3x2"
float="inline"
edges="left"
coords="none"
contents="R a2 1:b3"
/>
P2: <hexboard size="2x6"
float="inline"
edges="bottom"
coords="none"
contents="B c2 R 1:e1 E *:b1"
/>
P3: <hexboard size="3x5"
float="inline"
edges="bottom"
coords="none"
contents="B c3 R 1:d1 E *:(b2 e2)"
/>

===Making inferences: An extended example===

Use the patterns and intuitions that you know, to reason and make inferences in unfamiliar positions. You might not always find the best move, but you can often prune bad moves. Here is an extended example, involving several closely related patterns that are useful to know in isolation.
# ''Pattern:'' If you play the 4-4 3-3 obtuse corner joseki, it is important to know that Blue 6 [https://hexworld.org/board/#19nc1,d16c17d17c18d18e14c15 here] is weak.
#* This is because Red 7 undermines Blue 6 and also reduces Blue 2/4 from a 3rd row ladder escape to a 2nd row escape. Also, Blue's potential intrusion at c16 doesn't gain her anything.
#* Instead, [https://hexworld.org/board/#19nc1,d16c17d17c18d18d15 here] is a good local response for Blue. This gives Blue a 5th row ladder escape while blocking Red 1.
#* On the other hand, if Blue only has a 2nd row escape and doesn't have the "potential energy" to get a 3rd row escape, you might guess that Blue 6 above becomes a good local move, because Red 7 loses one of its main strengths. Such a guess would be correct: in [https://hexworld.org/board/#19nc1,d16c17p15d17e16g16e17f18d18e14c15 this position], Blue 10 is a strong move. On the other hand, Blue 10 at d15 would be weak here, as it's quite wasteful, only converting a 2nd row escape into a 3rd row escape.
# ''Pattern:'' If Red tenukis on move 3, then Blue 4 is locally strong in [https://hexworld.org/board/#19nc1,d16e14:pf15 this position]. In particular, Blue 4 is much stronger than bridging directly towards Blue's own edge with g13, because the former is a strong blocking move that reduces the effectiveness of Red 1. This strength is in spite of Red's intrusion on move 5. (I first saw this move in [https://www.littlegolem.net/jsp/game/game.jsp?gid=2090091&nmove=16 this bot game] from [https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=739 this thread on LittleGolem].)
#* Variations of this pattern appear frequently in obtuse corner joseki. If Red started with the 5-5 obtuse corner, then the equivalent Blue 4 is [https://hexworld.org/board/#19nc1,e15f13:pg14 also] strong.
#* Another simple example is Red 5 in [https://hexworld.org/board/#19nc1,d16d15f14:pe13 this position].
# ''Inference:'' Let us consider [https://hexworld.org/board/#19nc1,d16c17d17c18d18:p this position] again. Suppose Blue tenukis on move 6. What is a good local response for Red on move 7? There is more than one right answer, but you just need to find one move that isn't a blunder.
#* ''Hint:'' Use the two patterns above to come up with an answer.
#* ''Hint:'' Consider that Blue has a 3rd row ladder escape, so Blue d15 is locally strong but Blue e14 is weak. This is the case for future blue stones in this local region, even though Blue already played elsewhere on move 6. Therefore, Red should perhaps try to play a move that makes Blue d15 less attractive, even if it comes at the concession of making Blue e14 more attractive (since the latter is ''a priori'' a weak move, such a concession gives up less than it otherwise might).
#* ''Answer:'' Red 7 [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13 here] is a good local move.
#** Why? Ignoring stones 2-5, the presence of Red 1 and 7 makes Blue e14 more attractive. Red's 4-4 stone, much like the obtuse corner opening, tends to shift Blue's efficient stone locations up one row. With that in mind, Red's move 7 is a bit like Red e14 on an empty board, where Blue e15 is [https://hexworld.org/board/#19nc1,e14e15 the best response]. However, stones 2-5 make Blue e14 a bad move, despite Red 7 making it slightly less bad (this is the concession-but-not-really-a-concession that Red intended).
#** Let's now consider Blue d15. Normally a decent local move after stones 1-5, it becomes quite weak after Red 7, because of [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13d15f14 this sequence]. This is a typical example of the second pattern in this extended example. (Actually, if Blue had two moves in a row in this region, Blue 8 wouldn't be so bad because she could follow up with f14, where Red wants to play move 9. This is a tactical situation where it's bad for Red to tenuki.)
#** If we wanted to be complete, Blue actually has another option for move 8, f14 first which threatens d15: [https://hexworld.org/board/#19nc1,d16c17d17c18d18:pe13f14d15d14e14 here]. This is an example of the "skew cut" from above. Red should respond with d15 himself, or else Blue can play d15. This is probably Blue's best local response.
#** Every move makes a concession, but some concessions are more effective than others. As alluded to in the hint, Red 7 weakens a strong Blue reply (d15) while strengthening a weak Blue reply (e14). Red's goal is to minimize the strength of Blue's ''best'' response (rather than the strength of Blue's average response), and Red 7 at e13 accomplishes that beautifully.

Part of getting stronger at Hex (as with other games) is being able to compress knowledge and patterns effectively. If you understand this extended example, you can infer the best move in many similar but novel situations, without memorizing each one individually. If you forget which move is exactly the best, you can recover the right move with high probability by just reasoning about related patterns that you do know are strong or weak. The "A3 escape trick" example earlier in this article is another instance of this concept.

On larger boards and particularly near the opening or middlegame, when you have a lot of time to think about a position, it may be worth spending a good chunk of it making inferences and scoping out moves that are either good or to be avoided by each side, rather than spending most of your time calculating tons of lines and variations.
* In fact, once you've scoped out that "moves A and B are good for Red, while moves X and Y are bad for Red" and similarly for Blue, that makes the process of analyzing and pruning lines much more efficient.

==Joseki==

* The 4-3 corner move is almost always a mistake compared to the 5-4, even on 11×11. If your opponent plays 4-3, 90% of the time the best local reply is 3-3. If 4-3 is played, here are some common sequences:
<hexboard size="5x7"
coords="hide"
edges="bottom right"
contents="R 1:e2 B 2:e3 R 3:d3"
/>

<hexboard size="5x7"
coords="hide"
edges="bottom right"
contents="R 1:e2 B 2:e3 R 3:f2 B 4:f3 R 5:d3 E a:b4 b:c5"
/>

In the second joseki, Blue should play either (a) or (b), or she can defer the question until later. On larger boards with no nearby stones, (b) is often better.

* If your opponent plays the 5-4 joseki, a reasonable choice is to always play the 4-4 "high intrusion." KataHex prefers the 4-4 response by far, especially on 13×13 and larger. It's easily the "safest" choice, and many strong players have recently shown a clear preference for 4-4.

* In [https://hexworld.org/board/#14nc1,k10k11l10l12m11m12j11i13h12g14f13g12g13 this often-played joseki], Red 7 is a mistake because of the territory Blue gets after move 8.
** If the obtuse corner on Red's same edge is empty, a more principled response for Red is to play in that corner, like [https://hexworld.org/board/#14c1,k10k11l10l12d11 this].
** If the obtuse corner is already occupied by Blue, then Red should still respond in the obtuse corner. One reasonable option (among others) is [https://hexworld.org/board/#14c1,:pd11k10k11l10l12e11 this].

* See more discussion of acute corner joseki at [[Openings_on_19_x_19#Acute_corner_theory]]. For the obtuse corner, see [[Openings_on_19_x_19#Obtuse_corner_theory]] which covers responses to the 4-4 and 5-5 obtuse corner in great detail. Most of this is applicable to 13×13 and larger, and not just 19×19.

==Specific tactics==
===Threatening a bridge cut===
One of my favorite tactics is threatening a bridge cut, where the opponent needs to play too close to their own edge to defend:
<hexboard size="8x8"
float="inline"
edges="bottom left"
coords="hide"
contents="B *:d5 g2 R 1:f4 B 2:e3"
/>
Red 1 is a strong move, because it forces Blue to respond at 2 (or else Red will play the bridge cut there), but Blue 2 is normally overplaying on the left side when combined with Blue's stone at (*). Be on the lookout for this when your opponent plays the 4-4 and 7-7 obtuse corner, or 5-5 and 8-8.

This tactic can work well in other configurations, like [https://hexworld.org/board/#19c1,:pg8:pg11h9 this] and [https://hexworld.org/board/#19c1,:pe9:ph9g8 this]. However, be careful when using this tactic! When Blue isn't actually making a concession by bridging her two stones, like when the pattern occurs further from her own edge, it can be a blunder for Red to threaten a bridge cut. [https://hexworld.org/board/#19nc1,:pg8:pj5i7h6 Here] is an example where Red 5 is a blunder in the absence of nearby stones.

[https://www.littlegolem.net/jsp/game/game.jsp?gid=2436000&nmove=12 Here] is an example from a real game. Black's best move is 13. h5 (in the actual game, Black doesn't play this until move 19 but is still arguably rewarded for doing so). One might think that h5 is too strong near the top edge because of c2 and b5, but this is not the case because the white stone on c5 partially neutralizes b5.

[[category:strategy]]