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User:Hexanna

2026-07-20T15:36:31Z

Demer: /* 2-move openings for Bent-27 */

[[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 GUIs — 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 extensively when developing the study tools below.

[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 6-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.

==basic 1-2* hex strategy==

Here is a sample game of Hex with 1-2* turn order.

<hexboard size="11x11"
contents="R 1:j3
B 2:(i2 g2)
R 3:(j5 c5)
B 4:(c10 e10)
R 5:(d9 c9)
B 6:(e9 e8)
R 7:(c7 d10)
B 8:(d5 c6)
R 9:(i5 i7)
B 10:(h6 h5)
R 11:(i3 j7)
B 12:(e5 i9)
R 13:(d4 g9)
B 14:(b6 i6)
R 15:(j6 h3)
B 16:(h9 h8)
R 17:(h7 g8)
B 18:(f10 g10)
R 19:(e7 d8)
B 20:(h2 f7)
R 21:(g4 f6)
B 22:(f4 g7)
R 23:(f5 k9)
B 24:(b10 i10)
R 25:(d11 f3)
B 26:(g3 j2)
R 27:(k2 k1)"
/>

A few things stand out compared to regular Hex:
* In regular Hex, a bridge connects two stones. In 1-2* Hex, two stones like this are connected:
: <hexboard size="3x3" float="inline" coords="hide" edges="hide" visible="-(a1 c3)" contents="R a2 c2 E A:(b1 c1) B:(a3 b3) C:b2" />
: Red can play both A cells, both B cells, or C to connect. Blue cannot block all three threats in a two-stone turn.
* The corner created by moves 1 and 2 is interesting. Each player can connect at the corner but needs to spend ''two'' stones in order to do so. Any single move marked (*) allows the opponent to connect (with two stones).
: <hexboard size="3x3" float="inline" coords="hide" edges="top right" visible="-(a1 a3 c3)" contents="R 1:b3 B 2:a2 E *:(b1 c1 b2 c2)" />
* In regular Hex, the a2-b3-c4 diagonal is strong for Red. In 1-2* Hex, the a3-b4-c5 diagonal appears strong for Red. See 3. c5 or 23. k9, for example.
* Sometimes, the game partially collapses to single-turn Hex, where one side plays a move that would be an irrelevant but forcing move in regular Hex (like a [[time tesuji]]). A player with the initiative could use time tesujis to decide the tempo of play, choosing between one or two real moves per turn.

==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 are 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 a 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?]

==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 in the acute corner, 5-5 or 4-4 is a good choice on smaller boards. (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 distance 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, suggests to me that it is probably stronger than leela_bot and gzero_bot. I can't know for sure, though.
** On the other hand, the author seems to agree that 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).

==2-move openings for Bent-27==

question from [[User:Demer|Demer]]:
 
Have you tried finding a 2-move opening on the Bent-27
 
https://www.hexwiki.net/index.php/Where_to_swap_(y)#Bent-27
 
for which that modification of KataHex is not very confident of the winner?
 
(The first move would have to be one it thinks loses, since
 
winning first moves necessarily beat _every_ second move.)

hexanna: There are none. With enough visits (around 25k for the closest two-move openings), for each losing Red first move, each Blue reply is either <=7% or >=93% in winrate.

Thankyou for checking. [[User:Demer|Demer]]

User:Hexanna

2026-07-19T20:25:47Z

Demer:

Tom's move

2025-11-29T16:16:02Z

Demer: /* Bridge-first variant */ added other try by Blue

== Introduction ==

'''Tom's move''' is a trick that enables a player to make a connection from a 2nd-and-4th row [[parallel ladder]]. It can also be used to connect a 2nd row [[ladder]] using a single stone on the 4th row, or to connect a single stone on the 4th row to the edge. Its name originates from Tom Ace (player [[User:Tom239|Tom239]]), who devised it during a game against dj11, on 15 December 2002 on [[Playsite]]. This was not its first use ever, but it is how it came to be known among Hex players on Playsite.

== Description ==

Suppose Red has a 2nd-and-4th row [[parallel ladder]] and the amount of space shown here:
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 B b3 a5 E *:d3"
/>
Then Red can connect by playing at "*", the so-called "Tom's move".

== Usage examples ==

=== Connecting a 2nd row ladder using an isolated stone on the 4th row ===

Red to move and win:

<hexboard size="6x9"
edges="all"
coords="none"
contents="R a5 a4 d3 b2 B a6 b4 c3 d2 e1 f1 g1 h1 i1 h2 i2 i3"
/>

The solution is to [[ladder handling|push]] the [[ladder]] to 3 and then play Tom's move:

<hexboard size="6x9"
edges="all"
coords="none"
contents="R a5 a4 d3 b2 B a6 b4 c3 d2 e1 f1 g1 h1 i1 h2 i2 i3 R 1:b5 B 2:b6 R 3:c5 B 4:c6 R 5:f4"
/>

=== A single stone on the 4th row is connected ===

Consider a single stone on the 4th row, with the amount of space shown:
<hexboard size="5x9"
edges="bottom"
coords="none"
visible="-area(a1,a4,d1) h1 i1 i2"
contents="R arrow(12):d2"
/>

Then Red can connect as follows:
<hexboard size="5x9"
edges="bottom"
coords="none"
visible="-area(a1,a4,d1) h1 i1 i2"
contents="R arrow(12):d2 B 1:d3 R 2:c3 B 3:b5 R 4:c4 B 5:c5 R 6:f3"
/>
Red squeezes through the [[bottleneck]] at 2, starts a 2nd row ladder at 4, then plays Tom's move at 6. Note that all of Blue's moves are forced; if Blue plays differently, Red connects outright. This is [[edge template IV1d]].

=== In a game ===
Red to move:
<hexboard size="11x11"
edges="all"
coords="show"
contents="R b7 c6 d4 d5 d6 d8 f5 g3 g5 h5 i4 i6 j5 k2 k3 B c5 c7 d7 e4 e5 e6 f6 g6 h4 h6 h7 i3 i7 j3 j4 j6"
/>
Red's d4 [[group]] is already connected to the top edge by [[edge template IV1a|edge template IV1-a]]. To connect to the bottom, Red plays as follows:
<hexboard size="11x11"
edges="all"
coords="show"
contents="R b7 c6 d4 d5 d6 d8 f5 g3 g5 h5 i4 i6 j5 k2 k3 B c5 c7 d7 e4 e5 e6 f6 g6 h4 h6 h7 i3 i7 j3 j4 j6
R 1:b8 B 2:c9 R 3:a10 B 4:a11 R 5:b10 B 6:b11 R 7:c10 B 8:c11 R 9:f9"
/>
Now Red is connected by Tom's move. Note that d8 is already connected to Red's group by [[double threat]] at c8 and d9.

== Why Tom's move is connected ==

To compute Blue's [[mustplay region]], we consider two red [[threat]]s:

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:b4 4:c4 6:e4 B b3 a5 3:b5 5:c5 S b4 area(b5,d3,e3,e5)"
/>

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:c2 B b3 a5 S c2 c3 d2 area(b5,d3,e3,e5)"
/>

These leaves only blue moves in the [[ziggurat]].

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 B b3 a5 S area(b5,d3,e3,e5) E a:c4"
/>

If Blue plays there other than at a, then Red plays at a.
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:c4 B b3 a5 E *:c2 *:b4 z:b5 y:c5 x:(e5 d5 e4 d4 e3) S area(b5,d3,e3,e5)"
/>
In that case, Red's 2 connects back via either of the cells marked "*", and since the piece Blue just played is in only one of the x,y,z regions, Red's 2 also connects down via either of the remaining two of those three regions.

Thus Blue's only remaining hope is to play at a.

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 B b3 a5 1:c4"
/>

Red responds like this:

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:b4 4:e2 B b3 a5 1:c4 3:b5 E *:d1 *:c3"
/>

Red's 4 is now connected to the bottom via [[Edge_template_IV2b|edge template IV2b]], and to
Red's main group by double threat at the cells marked "*". Note that 2 and 3 do not actually need to be played; these moves have been included for clarity.

== Pushing the 4th row ladder first ==

Sometimes, there is not enough room to play Tom's move right away, but enough room can be created by first pushing the 4th row ladder. For a typical example, consider the following situation. It is Blue's turn, and Red wants to connect her stone to the bottom edge.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="-area(a1,a5,e1)"
contents="R d3 B e2 f2 j2 k3 g1--j1"
/>
If Red tries to play Tom's move immediately, it doesn't work, because 8 does not connect back to Red's main group.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="-area(a1,a5,e1)"
contents="R d3 B e2 f2 j2 k3 g1--j1 B 1:d4 R 2:c4 B 3:b6 R 4:c5 B 5:c6 R 6:f4 B 7:e5 R 8:g3 B 9:e4"
/>
What Red can do instead is start by pushing the 4th row ladder twice.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="-area(a1,a5,e1)"
contents="R d3 B e2 f2 j2 k3 g1--j1 B 1:d4 R 2:e3 B 3:e4 R 4:f3 B 5:f4 R 6:c4 B 7:b6 R 8:c5 B 9:c6 R 10:d5 B 11:d6 R 12:e5 B 13:e6 R 14:h4 B 15:g5 R 16:i3 E *:g4 *:h2"
/>
Of course this only works if after pushing the ladder, there is still enough room for Tom's move.

It is not actually necessary to push the 2nd row ladder (moves 6–13 can be omitted), but they have been included for clarity.

Note that when Red pushes the 4th row ladder, Blue cannot [[ladder handling|yield]], as this would give Red a [[ladder escape fork]] for the below 2nd row ladder. Also, as explained in more detail in the article on [[parallel ladder]]s, the 4th row ladder must be pushed ''before'' the 2nd row ladder has caught up to it. If Red starts by first pushing the 2nd row ladder, then it is too late to push the 4th row ladder.

== Variants ==

=== Alternative connection up ===

Tom's move also works when the hex marked "a" is not empty, provided that "b" connects to Red's main group.
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 B b3 a5 E a:c1 b:d1"
/>

For example, Tom's move works in this situation:
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(a1 b1 a2 a3 e1 f1 g1 f2 g2 g3)"
contents="R arrow(12):a4 a5 b3 B a6 b4 R d4 E b:d2 R c1 B a:c2 R b2"
/>

This is a special case of a [[Theorems_about_templates#Alternative_connection_up|general theorem]].

=== Tall variant ===

If "d" is empty, there is a variant of Tom's move that does not require a connection via "b", or even for "b" to be empty; it merely requires "c" and "e" to threaten to connect to Red's main group. An example is this situation:
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(f1 g1 g2 g3 area(a1,a3,c1))"
contents="R arrow(12):a4 a5 b3 B a6 b4 R d4 E c:e2 d:f2 e:d3 R d1 B b:d2 R c2"
/>
In this version of Tom's move, Red's 4 is different than usual (the usual move 4 does not work here). As before, moves 2 and 3 don't need to be played, but make it easier to see how the connection works.
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(f1 g1 g2 g3 area(a1,a3,c1))"
contents="R arrow(12):a4 a5 b3 B a6 b4 R x:d4 R d1 B d2 R c2
B 1:c5 R 2:b5 B 3:b6 R 4:e2 E y:e3"
/>
Note that "x" is connected to Red's main group without requiring "y", and "4" is also connected to Red's main group without requiring "y". (However, Red cannot guarantee to connect both "x" and "4" to her main group without requiring "y"). If Blue tries to cut Red off from the edge, Red responds as follows:
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(f1 g1 g2 g3 area(a1,a3,c1))"
contents="R arrow(12):a4 a5 b3 B a6 b4 R x:d4 R d1 B d2 R c2
B 1:c5 R 2:b5 B 3:b6 R 4:e2 E y:e3 B 5:d5 R 6:f4"
/>
Now no matter how Blue plays, Red can connect to the edge by a sequence of forcing moves. The situation is analogous to the [[Interior_template#The_hammock|hammock template]].

=== Bridge-first variant ===
If the end of Red's second row ladder is not yet directly beneath the end of Red's 4th row ladder, Red can opt to play the bridge to "1" first, instead of playing at "*".
<hexboard size="5x8"
edges="bottom"
coords="none"
visible="-(a1--a4 b2 b1 c1 g1 h1 h2)"
contents="R arrow(12):c2 b3 e3 1:e1 B c3 a5 d4 E *:f2"
/>

Unless Blue themselves plays at "*", Red can respond to any intrusion Blue makes by playing at "*". This will result either in Red being connected outright via the bridges and [[Edge_template_IV2b|edge template IV2b]], connecting with a double threat similar to the normal Tom's move, or connecting through a variation of the following double threat:

<hexboard size="5x8"
edges="bottom"
coords="none"
visible="-(a1--a4 b2 b1 c1 g1 h1 h2)"
contents="R arrow(12):c2 b3 e3 e1 2:f2 4:c4 B c3 a5 d4 1:e4 3:f4 S b4 d3"
/>

If Blue plays at "*", Red responds as follows, and has a double threat similar the normal Tom's move.
<hexboard size="5x8"
edges="bottom"
coords="none"
visible="-(a1--a4 b2 b1 c1 g1 h1 h2)"
contents="R arrow(12):c2 b3 e3 e1 2:e4 B c3 a5 d4 1:f2"
/>

This variation should be used if you don't want the opponent to get the territory from intruding into the bridge of a standard Tom's move. But what if the opponent plays that move before you do to ensure they get it? In this case it is often good to respond as follows, and use the tall variant to connect 2.

<hexboard size="6x8"
edges="bottom"
coords="none"
visible="-area(a1,a4,d1) a5 h2 h3 g1--h1"
contents="R arrow(12):c3 b4 e4 2:e1 B c4 a6 1:e2"
/>

Here is a specific example position where the bridge-first variant is necessary. Red's only winning move is Tom's move (a), and if Blue plays at (b), Red's only winning response is (c).
<hexboard size="7x7"
contents="R d1 b3 b4 a5 B b2 e1--g1 b5 E a:d5 b:c6 c:d3 d:d2"
/>

Furthermore, if just after Red plays Tom's move (a), Blue plays at (c) instead of (b), then Red's only winning response is (d).

== Tom's move for 3rd-and-5th row parallel ladders ==

Main article: [[Tom's move for 3rd and 5th row parallel ladders]].

There is a version of Tom's move that works for parallel ladders on the 3rd and 5th rows. It requires a large amount of space:
<hexboard size="6x12"
coords="hide"
edges="bottom"
visible="area(d1,b3,b6,l6,l4,j2,f1)"
contents="R arrow(12):c2 b3 b4 B b5 c3 R 1:e3"
/>
By playing at "1", Red can connect to the edge. Verifying this requires a lot of steps, but here is the basic idea:
<hexboard size="6x12"
coords="hide"
edges="bottom"
visible="area(d1,b3,b6,l6,l4,j2,f1)"
contents="R arrow(12):c2 b3 b4 B b5 c3 E x:i4
R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1"
/>
Notice that Red's 3 is connected left by double threat at the two cells marked "*", and connected right by [[Fifth_row_edge_templates#V-2-m|edge template V2m]]. The latter template is itself based on Tom's move at "x". It works, for example, like this:
<hexboard size="6x12"
coords="hide"
edges="bottom"
visible="area(d1,b3,b6,l6,l4,j2,f1)"
contents="R arrow(12):c2 b3 b4 B b5 c3
R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1
B 6:f4 R 7:g3 B 8:g4 R 9:e4 B 10:d6 R 11:e5 B 12:e6 R 13:f5 B 14:f6 R 15:i4"
/>
Now Red is connected by the (ordinary) Tom's move.

See [[Tom's move for 3rd and 5th row parallel ladders]] for more details.

=== Tall variant ===

Tom's move for 3rd-and-5th row parallel ladders also has a tall version. It works in the same way as the tall version of Tom's move for 2nd-and-4th row parallel ladders above, given the appropriate amount of space.

<hexboard size="7x12"
coords="hide"
edges="bottom"
visible="area(d2,b4,b7,l7,l5,j3,g2,f1,e1)"
contents="R arrow(12):e1 d2 c3 b4 b5 B b6 c4 e2 R 1:e4 B 2:d5 R 3:f2"
/>

== See also ==

* [[Parallel ladder]]
* [[Edge template IV1d]]
* [[Fifth_row_edge_templates#V-2-m|Edge template V2m]]

[[category:ladder]]
[[category:Advanced Strategy]]

Tom's move

2025-11-27T22:07:14Z

Demer: /* Bridge-first variant */ fixed cell reference

== Introduction ==

'''Tom's move''' is a trick that enables a player to make a connection from a 2nd-and-4th row [[parallel ladder]]. It can also be used to connect a 2nd row [[ladder]] using a single stone on the 4th row, or to connect a single stone on the 4th row to the edge. Its name originates from Tom Ace (player [[User:Tom239|Tom239]]), who devised it during a game against dj11, on 15 December 2002 on [[Playsite]]. This was not its first use ever, but it is how it came to be known among Hex players on Playsite.

== Description ==

Suppose Red has a 2nd-and-4th row [[parallel ladder]] and the amount of space shown here:
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 B b3 a5 E *:d3"
/>
Then Red can connect by playing at "*", the so-called "Tom's move".

== Usage examples ==

=== Connecting a 2nd row ladder using an isolated stone on the 4th row ===

Red to move and win:

<hexboard size="6x9"
edges="all"
coords="none"
contents="R a5 a4 d3 b2 B a6 b4 c3 d2 e1 f1 g1 h1 i1 h2 i2 i3"
/>

The solution is to [[ladder handling|push]] the [[ladder]] to 3 and then play Tom's move:

<hexboard size="6x9"
edges="all"
coords="none"
contents="R a5 a4 d3 b2 B a6 b4 c3 d2 e1 f1 g1 h1 i1 h2 i2 i3 R 1:b5 B 2:b6 R 3:c5 B 4:c6 R 5:f4"
/>

=== A single stone on the 4th row is connected ===

Consider a single stone on the 4th row, with the amount of space shown:
<hexboard size="5x9"
edges="bottom"
coords="none"
visible="-area(a1,a4,d1) h1 i1 i2"
contents="R arrow(12):d2"
/>

Then Red can connect as follows:
<hexboard size="5x9"
edges="bottom"
coords="none"
visible="-area(a1,a4,d1) h1 i1 i2"
contents="R arrow(12):d2 B 1:d3 R 2:c3 B 3:b5 R 4:c4 B 5:c5 R 6:f3"
/>
Red squeezes through the [[bottleneck]] at 2, starts a 2nd row ladder at 4, then plays Tom's move at 6. Note that all of Blue's moves are forced; if Blue plays differently, Red connects outright. This is [[edge template IV1d]].

=== In a game ===
Red to move:
<hexboard size="11x11"
edges="all"
coords="show"
contents="R b7 c6 d4 d5 d6 d8 f5 g3 g5 h5 i4 i6 j5 k2 k3 B c5 c7 d7 e4 e5 e6 f6 g6 h4 h6 h7 i3 i7 j3 j4 j6"
/>
Red's d4 [[group]] is already connected to the top edge by [[edge template IV1a|edge template IV1-a]]. To connect to the bottom, Red plays as follows:
<hexboard size="11x11"
edges="all"
coords="show"
contents="R b7 c6 d4 d5 d6 d8 f5 g3 g5 h5 i4 i6 j5 k2 k3 B c5 c7 d7 e4 e5 e6 f6 g6 h4 h6 h7 i3 i7 j3 j4 j6
R 1:b8 B 2:c9 R 3:a10 B 4:a11 R 5:b10 B 6:b11 R 7:c10 B 8:c11 R 9:f9"
/>
Now Red is connected by Tom's move. Note that d8 is already connected to Red's group by [[double threat]] at c8 and d9.

== Why Tom's move is connected ==

To compute Blue's [[mustplay region]], we consider two red [[threat]]s:

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:b4 4:c4 6:e4 B b3 a5 3:b5 5:c5 S b4 area(b5,d3,e3,e5)"
/>

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:c2 B b3 a5 S c2 c3 d2 area(b5,d3,e3,e5)"
/>

These leaves only blue moves in the [[ziggurat]].

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 B b3 a5 S area(b5,d3,e3,e5) E a:c4"
/>

If Blue plays there other than at a, then Red plays at a.
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:c4 B b3 a5 E *:c2 *:b4 z:b5 y:c5 x:(e5 d5 e4 d4 e3) S area(b5,d3,e3,e5)"
/>
In that case, Red's 2 connects back via either of the cells marked "*", and since the piece Blue just played is in only one of the x,y,z regions, Red's 2 also connects down via either of the remaining two of those three regions.

Thus Blue's only remaining hope is to play at a.

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 B b3 a5 1:c4"
/>

Red responds like this:

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:b4 4:e2 B b3 a5 1:c4 3:b5 E *:d1 *:c3"
/>

Red's 4 is now connected to the bottom via [[Edge_template_IV2b|edge template IV2b]], and to
Red's main group by double threat at the cells marked "*". Note that 2 and 3 do not actually need to be played; these moves have been included for clarity.

== Pushing the 4th row ladder first ==

Sometimes, there is not enough room to play Tom's move right away, but enough room can be created by first pushing the 4th row ladder. For a typical example, consider the following situation. It is Blue's turn, and Red wants to connect her stone to the bottom edge.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="-area(a1,a5,e1)"
contents="R d3 B e2 f2 j2 k3 g1--j1"
/>
If Red tries to play Tom's move immediately, it doesn't work, because 8 does not connect back to Red's main group.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="-area(a1,a5,e1)"
contents="R d3 B e2 f2 j2 k3 g1--j1 B 1:d4 R 2:c4 B 3:b6 R 4:c5 B 5:c6 R 6:f4 B 7:e5 R 8:g3 B 9:e4"
/>
What Red can do instead is start by pushing the 4th row ladder twice.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="-area(a1,a5,e1)"
contents="R d3 B e2 f2 j2 k3 g1--j1 B 1:d4 R 2:e3 B 3:e4 R 4:f3 B 5:f4 R 6:c4 B 7:b6 R 8:c5 B 9:c6 R 10:d5 B 11:d6 R 12:e5 B 13:e6 R 14:h4 B 15:g5 R 16:i3 E *:g4 *:h2"
/>
Of course this only works if after pushing the ladder, there is still enough room for Tom's move.

It is not actually necessary to push the 2nd row ladder (moves 6–13 can be omitted), but they have been included for clarity.

Note that when Red pushes the 4th row ladder, Blue cannot [[ladder handling|yield]], as this would give Red a [[ladder escape fork]] for the below 2nd row ladder. Also, as explained in more detail in the article on [[parallel ladder]]s, the 4th row ladder must be pushed ''before'' the 2nd row ladder has caught up to it. If Red starts by first pushing the 2nd row ladder, then it is too late to push the 4th row ladder.

== Variants ==

=== Alternative connection up ===

Tom's move also works when the hex marked "a" is not empty, provided that "b" connects to Red's main group.
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 B b3 a5 E a:c1 b:d1"
/>

For example, Tom's move works in this situation:
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(a1 b1 a2 a3 e1 f1 g1 f2 g2 g3)"
contents="R arrow(12):a4 a5 b3 B a6 b4 R d4 E b:d2 R c1 B a:c2 R b2"
/>

This is a special case of a [[Theorems_about_templates#Alternative_connection_up|general theorem]].

=== Tall variant ===

If "d" is empty, there is a variant of Tom's move that does not require a connection via "b", or even for "b" to be empty; it merely requires "c" and "e" to threaten to connect to Red's main group. An example is this situation:
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(f1 g1 g2 g3 area(a1,a3,c1))"
contents="R arrow(12):a4 a5 b3 B a6 b4 R d4 E c:e2 d:f2 e:d3 R d1 B b:d2 R c2"
/>
In this version of Tom's move, Red's 4 is different than usual (the usual move 4 does not work here). As before, moves 2 and 3 don't need to be played, but make it easier to see how the connection works.
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(f1 g1 g2 g3 area(a1,a3,c1))"
contents="R arrow(12):a4 a5 b3 B a6 b4 R x:d4 R d1 B d2 R c2
B 1:c5 R 2:b5 B 3:b6 R 4:e2 E y:e3"
/>
Note that "x" is connected to Red's main group without requiring "y", and "4" is also connected to Red's main group without requiring "y". (However, Red cannot guarantee to connect both "x" and "4" to her main group without requiring "y"). If Blue tries to cut Red off from the edge, Red responds as follows:
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(f1 g1 g2 g3 area(a1,a3,c1))"
contents="R arrow(12):a4 a5 b3 B a6 b4 R x:d4 R d1 B d2 R c2
B 1:c5 R 2:b5 B 3:b6 R 4:e2 E y:e3 B 5:d5 R 6:f4"
/>
Now no matter how Blue plays, Red can connect to the edge by a sequence of forcing moves. The situation is analogous to the [[Interior_template#The_hammock|hammock template]].

=== Bridge-first variant ===
If the end of Red's second row ladder is not yet directly beneath the end of Red's 4th row ladder, Red can opt to play the bridge to "1" first, instead of playing at "*".
<hexboard size="5x8"
edges="bottom"
coords="none"
visible="-(a1--a4 b2 b1 c1 g1 h1 h2)"
contents="R arrow(12):c2 b3 e3 1:e1 B c3 a5 d4 E *:f2"
/>

Unless Blue themselves plays at "*", Red can respond to any intrusion Blue makes by playing at "*". This will result either in Red being connected outright via the bridges and [[Edge_template_IV2b|edge template IV2b]], connecting with a double threat similar to the normal Tom's move, or connecting through a variation of the following double threat:

<hexboard size="5x8"
edges="bottom"
coords="none"
visible="-(a1--a4 b2 b1 c1 g1 h1 h2)"
contents="R arrow(12):c2 b3 e3 e1 2:f2 4:c4 B c3 a5 d4 1:e4 3:f4 S b4 d3"
/>

If Blue plays at "*", Red responds as follows, and has a double threat similar the normal Tom's move.
<hexboard size="5x8"
edges="bottom"
coords="none"
visible="-(a1--a4 b2 b1 c1 g1 h1 h2)"
contents="R arrow(12):c2 b3 e3 e1 2:e4 B c3 a5 d4 1:f2"
/>

This variation should be used if you don't want the opponent to get the territory from intruding into the bridge of a standard Tom's move. But what if the opponent plays that move before you do to ensure they get it? In this case it is often good to respond as follows, and use the tall variant to connect 2.

<hexboard size="6x8"
edges="bottom"
coords="none"
visible="-area(a1,a4,d1) a5 h2 h3 g1--h1"
contents="R arrow(12):c3 b4 e4 2:e1 B c4 a6 1:e2"
/>

Here is a specific example position where the bridge-first variant is necessary. Red's only winning move is Tom's move (a), and if Blue plays at (b), Red's only winning response is (c).
<hexboard size="7x7"
contents="R d1 b3 b4 a5 B b2 e1--g1 b5 E a:d5 b:c6 c:d3"
/>

== Tom's move for 3rd-and-5th row parallel ladders ==

Main article: [[Tom's move for 3rd and 5th row parallel ladders]].

There is a version of Tom's move that works for parallel ladders on the 3rd and 5th rows. It requires a large amount of space:
<hexboard size="6x12"
coords="hide"
edges="bottom"
visible="area(d1,b3,b6,l6,l4,j2,f1)"
contents="R arrow(12):c2 b3 b4 B b5 c3 R 1:e3"
/>
By playing at "1", Red can connect to the edge. Verifying this requires a lot of steps, but here is the basic idea:
<hexboard size="6x12"
coords="hide"
edges="bottom"
visible="area(d1,b3,b6,l6,l4,j2,f1)"
contents="R arrow(12):c2 b3 b4 B b5 c3 E x:i4
R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1"
/>
Notice that Red's 3 is connected left by double threat at the two cells marked "*", and connected right by [[Fifth_row_edge_templates#V-2-m|edge template V2m]]. The latter template is itself based on Tom's move at "x". It works, for example, like this:
<hexboard size="6x12"
coords="hide"
edges="bottom"
visible="area(d1,b3,b6,l6,l4,j2,f1)"
contents="R arrow(12):c2 b3 b4 B b5 c3
R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1
B 6:f4 R 7:g3 B 8:g4 R 9:e4 B 10:d6 R 11:e5 B 12:e6 R 13:f5 B 14:f6 R 15:i4"
/>
Now Red is connected by the (ordinary) Tom's move.

See [[Tom's move for 3rd and 5th row parallel ladders]] for more details.

=== Tall variant ===

Tom's move for 3rd-and-5th row parallel ladders also has a tall version. It works in the same way as the tall version of Tom's move for 2nd-and-4th row parallel ladders above, given the appropriate amount of space.

<hexboard size="7x12"
coords="hide"
edges="bottom"
visible="area(d2,b4,b7,l7,l5,j3,g2,f1,e1)"
contents="R arrow(12):e1 d2 c3 b4 b5 B b6 c4 e2 R 1:e4 B 2:d5 R 3:f2"
/>

== See also ==

* [[Parallel ladder]]
* [[Edge template IV1d]]
* [[Fifth_row_edge_templates#V-2-m|Edge template V2m]]

[[category:ladder]]
[[category:Advanced Strategy]]

Tom's move

2025-11-26T23:48:43Z

Demer: /* Variants */ removed unneeded cells and stones

== Introduction ==

'''Tom's move''' is a trick that enables a player to make a connection from a 2nd-and-4th row [[parallel ladder]]. It can also be used to connect a 2nd row [[ladder]] using a single stone on the 4th row, or to connect a single stone on the 4th row to the edge. Its name originates from Tom Ace (player [[User:Tom239|Tom239]]), who devised it during a game against dj11, on 15 December 2002 on [[Playsite]]. This was not its first use ever, but it is how it came to be known among Hex players on Playsite.

== Description ==

Suppose Red has a 2nd-and-4th row [[parallel ladder]] and the amount of space shown here:
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 B b3 a5 E *:d3"
/>
Then Red can connect by playing at "*", the so-called "Tom's move".

== Usage examples ==

=== Connecting a 2nd row ladder using an isolated stone on the 4th row ===

Red to move and win:

<hexboard size="6x9"
edges="all"
coords="none"
contents="R a5 a4 d3 b2 B a6 b4 c3 d2 e1 f1 g1 h1 i1 h2 i2 i3"
/>

The solution is to [[ladder handling|push]] the [[ladder]] to 3 and then play Tom's move:

<hexboard size="6x9"
edges="all"
coords="none"
contents="R a5 a4 d3 b2 B a6 b4 c3 d2 e1 f1 g1 h1 i1 h2 i2 i3 R 1:b5 B 2:b6 R 3:c5 B 4:c6 R 5:f4"
/>

=== A single stone on the 4th row is connected ===

Consider a single stone on the 4th row, with the amount of space shown:
<hexboard size="5x9"
edges="bottom"
coords="none"
visible="-area(a1,a4,d1) h1 i1 i2"
contents="R arrow(12):d2"
/>

Then Red can connect as follows:
<hexboard size="5x9"
edges="bottom"
coords="none"
visible="-area(a1,a4,d1) h1 i1 i2"
contents="R arrow(12):d2 B 1:d3 R 2:c3 B 3:b5 R 4:c4 B 5:c5 R 6:f3"
/>
Red squeezes through the [[bottleneck]] at 2, starts a 2nd row ladder at 4, then plays Tom's move at 6. Note that all of Blue's moves are forced; if Blue plays differently, Red connects outright. This is [[edge template IV1d]].

=== In a game ===
Red to move:
<hexboard size="11x11"
edges="all"
coords="show"
contents="R b7 c6 d4 d5 d6 d8 f5 g3 g5 h5 i4 i6 j5 k2 k3 B c5 c7 d7 e4 e5 e6 f6 g6 h4 h6 h7 i3 i7 j3 j4 j6"
/>
Red's d4 [[group]] is already connected to the top edge by [[edge template IV1a|edge template IV1-a]]. To connect to the bottom, Red plays as follows:
<hexboard size="11x11"
edges="all"
coords="show"
contents="R b7 c6 d4 d5 d6 d8 f5 g3 g5 h5 i4 i6 j5 k2 k3 B c5 c7 d7 e4 e5 e6 f6 g6 h4 h6 h7 i3 i7 j3 j4 j6
R 1:b8 B 2:c9 R 3:a10 B 4:a11 R 5:b10 B 6:b11 R 7:c10 B 8:c11 R 9:f9"
/>
Now Red is connected by Tom's move. Note that d8 is already connected to Red's group by [[double threat]] at c8 and d9.

== Why Tom's move is connected ==

To compute Blue's [[mustplay region]], we consider two red [[threat]]s:

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:b4 4:c4 6:e4 B b3 a5 3:b5 5:c5 S b4 area(b5,d3,e3,e5)"
/>

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:c2 B b3 a5 S c2 c3 d2 area(b5,d3,e3,e5)"
/>

These leaves only blue moves in the [[ziggurat]].

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 B b3 a5 S area(b5,d3,e3,e5) E a:c4"
/>

If Blue plays there other than at a, then Red plays at a.
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:c4 B b3 a5 E *:c2 *:b4 z:b5 y:c5 x:(e5 d5 e4 d4 e3) S area(b5,d3,e3,e5)"
/>
In that case, Red's 2 connects back via either of the cells marked "*", and since the piece Blue just played is in only one of the x,y,z regions, Red's 2 also connects down via either of the remaining two of those three regions.

Thus Blue's only remaining hope is to play at a.

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 B b3 a5 1:c4"
/>

Red responds like this:

<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 2:b4 4:e2 B b3 a5 1:c4 3:b5 E *:d1 *:c3"
/>

Red's 4 is now connected to the bottom via [[Edge_template_IV2b|edge template IV2b]], and to
Red's main group by double threat at the cells marked "*". Note that 2 and 3 do not actually need to be played; these moves have been included for clarity.

== Pushing the 4th row ladder first ==

Sometimes, there is not enough room to play Tom's move right away, but enough room can be created by first pushing the 4th row ladder. For a typical example, consider the following situation. It is Blue's turn, and Red wants to connect her stone to the bottom edge.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="-area(a1,a5,e1)"
contents="R d3 B e2 f2 j2 k3 g1--j1"
/>
If Red tries to play Tom's move immediately, it doesn't work, because 8 does not connect back to Red's main group.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="-area(a1,a5,e1)"
contents="R d3 B e2 f2 j2 k3 g1--j1 B 1:d4 R 2:c4 B 3:b6 R 4:c5 B 5:c6 R 6:f4 B 7:e5 R 8:g3 B 9:e4"
/>
What Red can do instead is start by pushing the 4th row ladder twice.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="-area(a1,a5,e1)"
contents="R d3 B e2 f2 j2 k3 g1--j1 B 1:d4 R 2:e3 B 3:e4 R 4:f3 B 5:f4 R 6:c4 B 7:b6 R 8:c5 B 9:c6 R 10:d5 B 11:d6 R 12:e5 B 13:e6 R 14:h4 B 15:g5 R 16:i3 E *:g4 *:h2"
/>
Of course this only works if after pushing the ladder, there is still enough room for Tom's move.

It is not actually necessary to push the 2nd row ladder (moves 6–13 can be omitted), but they have been included for clarity.

Note that when Red pushes the 4th row ladder, Blue cannot [[ladder handling|yield]], as this would give Red a [[ladder escape fork]] for the below 2nd row ladder. Also, as explained in more detail in the article on [[parallel ladder]]s, the 4th row ladder must be pushed ''before'' the 2nd row ladder has caught up to it. If Red starts by first pushing the 2nd row ladder, then it is too late to push the 4th row ladder.

== Variants ==

=== Alternative connection up ===

Tom's move also works when the hex marked "a" is not empty, provided that "b" connects to Red's main group.
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 d3 B b3 a5 E a:c1 b:d1"
/>

For example, Tom's move works in this situation:
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(a1 b1 a2 a3 e1 f1 g1 f2 g2 g3)"
contents="R arrow(12):a4 a5 b3 B a6 b4 R d4 E b:d2 R c1 B a:c2 R b2"
/>

This is a special case of a [[Theorems_about_templates#Alternative_connection_up|general theorem]].

=== Tall variant ===

If "d" is empty, there is a variant of Tom's move that does not require a connection via "b", or even for "b" to be empty; it merely requires "c" and "e" to threaten to connect to Red's main group. An example is this situation:
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(f1 g1 g2 g3 area(a1,a3,c1))"
contents="R arrow(12):a4 a5 b3 B a6 b4 R d4 E c:e2 d:f2 e:d3 R d1 B b:d2 R c2"
/>
In this version of Tom's move, Red's 4 is different than usual (the usual move 4 does not work here). As before, moves 2 and 3 don't need to be played, but make it easier to see how the connection works.
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(f1 g1 g2 g3 area(a1,a3,c1))"
contents="R arrow(12):a4 a5 b3 B a6 b4 R x:d4 R d1 B d2 R c2
B 1:c5 R 2:b5 B 3:b6 R 4:e2 E y:e3"
/>
Note that "x" is connected to Red's main group without requiring "y", and "4" is also connected to Red's main group without requiring "y". (However, Red cannot guarantee to connect both "x" and "4" to her main group without requiring "y"). If Blue tries to cut Red off from the edge, Red responds as follows:
<hexboard size="6x7"
edges="bottom"
coords="none"
visible="-(f1 g1 g2 g3 area(a1,a3,c1))"
contents="R arrow(12):a4 a5 b3 B a6 b4 R x:d4 R d1 B d2 R c2
B 1:c5 R 2:b5 B 3:b6 R 4:e2 E y:e3 B 5:d5 R 6:f4"
/>
Now no matter how Blue plays, Red can connect to the edge by a sequence of forcing moves. The situation is analogous to the [[Interior_template#The_hammock|hammock template]].

=== Bridge-first variant ===
If the end of Red's second row ladder is not yet directly beneath the end of their 4th row ladder, Red can opt to play the bridge to "*" first, instead of playing at "*".
<hexboard size="5x8"
edges="bottom"
coords="none"
visible="-(a1--a4 b2 b1 c1 g1 h1 h2)"
contents="R arrow(12):c2 b3 e3 1:e1 B c3 a5 d4 E *:f2"
/>

Unless Blue themselves plays at "*", Red can respond to any intrusion Blue makes by playing at "*". This will result either in Red being connected outright via the bridges and [[Edge_template_IV2b|edge template IV2b]], connecting with a double threat similar to the normal Tom's move, or connecting through a variation of the following double threat:

<hexboard size="5x8"
edges="bottom"
coords="none"
visible="-(a1--a4 b2 b1 c1 g1 h1 h2)"
contents="R arrow(12):c2 b3 e3 e1 2:f2 4:c4 B c3 a5 d4 1:e4 3:f4 S b4 d3"
/>

If Blue plays at "*", Red responds as follows, and has a double threat similar the normal Tom's move.
<hexboard size="5x8"
edges="bottom"
coords="none"
visible="-(a1--a4 b2 b1 c1 g1 h1 h2)"
contents="R arrow(12):c2 b3 e3 e1 2:e4 B c3 a5 d4 1:f2"
/>

This variation should be used if you don't want the opponent to get the territory from intruding into the bridge of a standard Tom's move. But what if the opponent plays that move before you do to ensure they get it? In this case it is often good to respond as follows, and use the tall variant to connect 2.

<hexboard size="6x8"
edges="bottom"
coords="none"
visible="-area(a1,a4,d1) a5 h2 h3 g1--h1"
contents="R arrow(12):c3 b4 e4 2:e1 B c4 a6 1:e2"
/>

Here is a specific example position where the bridge-first variant is necessary. Red's only winning move is Tom's move (a), and if Blue plays at (b), Red's only winning response is (c).
<hexboard size="7x7"
contents="R d1 b3 b4 a5 B b2 e1--g1 b5 E a:d5 b:c6 c:d3"
/>

== Tom's move for 3rd-and-5th row parallel ladders ==

Main article: [[Tom's move for 3rd and 5th row parallel ladders]].

There is a version of Tom's move that works for parallel ladders on the 3rd and 5th rows. It requires a large amount of space:
<hexboard size="6x12"
coords="hide"
edges="bottom"
visible="area(d1,b3,b6,l6,l4,j2,f1)"
contents="R arrow(12):c2 b3 b4 B b5 c3 R 1:e3"
/>
By playing at "1", Red can connect to the edge. Verifying this requires a lot of steps, but here is the basic idea:
<hexboard size="6x12"
coords="hide"
edges="bottom"
visible="area(d1,b3,b6,l6,l4,j2,f1)"
contents="R arrow(12):c2 b3 b4 B b5 c3 E x:i4
R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1"
/>
Notice that Red's 3 is connected left by double threat at the two cells marked "*", and connected right by [[Fifth_row_edge_templates#V-2-m|edge template V2m]]. The latter template is itself based on Tom's move at "x". It works, for example, like this:
<hexboard size="6x12"
coords="hide"
edges="bottom"
visible="area(d1,b3,b6,l6,l4,j2,f1)"
contents="R arrow(12):c2 b3 b4 B b5 c3
R 1:e3 B 2:d4 R 3:c4 B 4:c5 R 5:f2 E *:d3 *:e1
B 6:f4 R 7:g3 B 8:g4 R 9:e4 B 10:d6 R 11:e5 B 12:e6 R 13:f5 B 14:f6 R 15:i4"
/>
Now Red is connected by the (ordinary) Tom's move.

See [[Tom's move for 3rd and 5th row parallel ladders]] for more details.

=== Tall variant ===

Tom's move for 3rd-and-5th row parallel ladders also has a tall version. It works in the same way as the tall version of Tom's move for 2nd-and-4th row parallel ladders above, given the appropriate amount of space.

<hexboard size="7x12"
coords="hide"
edges="bottom"
visible="area(d2,b4,b7,l7,l5,j3,g2,f1,e1)"
contents="R arrow(12):e1 d2 c3 b4 b5 B b6 c4 e2 R 1:e4 B 2:d5 R 3:f2"
/>

== See also ==

* [[Parallel ladder]]
* [[Edge template IV1d]]
* [[Fifth_row_edge_templates#V-2-m|Edge template V2m]]

[[category:ladder]]
[[category:Advanced Strategy]]

Second order template

2025-08-30T23:57:13Z

Demer: /* Examples */ added fifth row template

A '''second order template''' is a [[template]] that guarantees a connection even if the opponent starts with two moves in the template. Put another way, a second order template is a pattern in which an intrusion is not a [[forcing move]]. A pattern can be proved to be a second order template by showing that every possible intrusion preserves at least one [[template|first order template]]. To qualify as a second order template, the pattern should also be minimal.

Phrases such as "Blue cannot even threaten to disconnect Red" and "if Blue moves in this area, Red can just ignore it" are often used to indicate that Red has a second order template.

== Examples ==
=== Second row ===
<hexboard size="2x3"
coords="none"
edges="bottom"
visible="-a1"
contents="R b1 c1"
/>
=== Third row ===
<hexboard size="3x6"
coords="none"
edges="bottom"
visible="-a1 b1 a2"
contents="R d1 e1"
/>

Red has ''three'' non-overlapping threats:

<hexboard size="3x6"
coords="none"
edges="bottom"
visible="-a1 b1 a2"
contents="R d1 e1 S a3 b3 b2 c2 c1 E *:b2"
/>

<hexboard size="3x6"
coords="none"
edges="bottom"
visible="-a1 b1 a2"
contents="R d1 e1 S c3 d2 d3 E *:d2"
/>

<hexboard size="3x6"
coords="none"
edges="bottom"
visible="-a1 b1 a2"
contents="R d1 e1 S f1 e2 f2 e3 f3 E *:f2"
/>

So no matter where Blue plays in the template, Red will still have at least two non-overlapping threats, and therefore a first-order template, remaining. Or to put it another way: If Blue makes two moves in the template, Blue can disable at most two of Red's threats, so Red can still use the third one to reconnect.

Here is another second order template on the third row:
<hexboard size="3x4"
coords="none"
edges="bottom"
visible="-a1 a2 b1"
contents="R c1 b2"
/>

=== Fourth row ===
The following is a second order template:
<hexboard size="4x6"
coords="none"
edges="bottom"
visible="-area(a1,a3,c1)"
contents="R d1 e1 f1"
/>

=== Fifth row ===
The following is a second order template:
<hexboard size="5x9"
coords="none"
edges="bottom"
visible="-area(a1,a4,d1)"
contents="R f1 g1 h1"
/>

== Usage ==

=== In play ===

It is usually not a good idea to create a second order template on purpose, as this tends to waste a move that would be better spent elsewhere. However, it is still useful to recognize second order templates in case they form accidentally.

When the opponent intrudes into a first order template, it is usually necessary to defend the template to preserve the connection. The opponent can take advantage of this by playing template intrusions that will later be useful to the opponent, for example as [[ladder escape]]s or to gain [[territory]]. Such moves belong to the category of [[double threat]]s.

On the other hand, when the opponent intrudes into a second order template, no immediate response is necessary; the template's owner can simply ignore the intrusion and is free to move elsewhere, thereby gaining the [[initiative]]. Recognizing second order templates helps to know whether an area is safe or might be subject to threats.

=== In mustplay analysis ===

Second order templates can sometimes be useful in the analysis of Hex positions, such as [[mustplay region|mustplay analysis]]. For example, suppose we want to prove the correctness of the following 6th row (first order) edge template:
<hexboard size="6x7"
coords="none"
edges="bottom"
visible="-area(a1,a5,e1) g1 g2"
contents="R f1 f3 g3 E a:e2 b:f2 c:e3"
/>
We can reason as follows: if Blue plays anywhere in the template except a, b, or c, then Red can play at c, forming the second order template
<hexboard size="4x6"
coords="none"
edges="bottom"
visible="-area(a1,a3,c1)"
contents="R d1 e1 f1"
/>
Since Blue has at most one stone in this template, the result is still a first-order template, so that Red is connected to the edge. Therefore, the only intrusions we need to consider are a, b, and c. This greatly simplifies the analysis, as we must now only consider 3 possible intrusions, rather than all 22 of them. (The intrusions at a or c are easily dealt with, since Red can simply respond at b to connect via [[edge template IV2a]]. The final intrusion at b has a few further cases to consider, but is relatively straighforward).

== Second order ladder creation templates ==

The following is a second order [[ladder creation template]].
<hexboard size="4x6"
coords="none"
edges="bottom"
visible="-area(a1,a3,c1)-f1"
contents="R d1 c2 S red:(b3 a4) E arrow(9):(b3 a4)"
/>
What this means is that even if Blue is allowed two moves in the template, Red can still guarantee to either connect to the edge or get the indicated 2nd row [[ladder]].

[[category:templates]]
[[category:connection types]]
[[category:advanced Strategy]]

Open problems

2025-03-15T12:13:56Z

Demer: showed that a generalization is false

* Are there cells other than a1 and b1 which are theoretically losing first moves?

* Is it true that for every cell (defined in terms of direction and distance from an [[Board#Corners|acute corner]]) there is an ''n'' such that for any [[Board]] of size at least ''n'' that cell is a losing [[opening|opening move]]?

* Conversely, is it true that, for example, c3 is a winning first move on every Hex board of size at least 5?

* Is the [[center opening|center hex]] on every Hex board of [[Board_size|odd size]] a winning opening move?

* On boards of all [[board size|sizes]], is every opening move on the [[Board#Diagonals|short diagonal]] winning?

* Is the following true? Assume one player is in a winning position (will win with [[optimal play]]) and the opponent plays in a hex X. Let the set A consist of all empty hexes that are members of any path between opponents edges that uses the stone at X. If A is non-empty, A contains a winning move. Otherwise any move is winning, even [[passing]] the turn. (This problem was posed by [[Jory]] in the [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=167 Little Golem forum].)

:*The generalization of the above statement to monotone set-coloring games is false, as shown in section 1.4 .

== Formerly open problems ==

=== [[Sixth row template problem]] ===

Does there exist an [[edge template]] which guarantees a secure [[connection]] for a [[piece]] on the sixth row?

'''Answer:''' Yes, [[edge template VI1a]] is such a template.

=== Triangle template problem ===

Are the templates below valid in their generalization to larger sizes? (This problem was posed by [[Jory]] in the [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=167 Little Golem forum].)

<hexboard size="1x1"
float="inline"
edges="bottom"
coords="none"
contents="R a1"/><hexboard size="2x2"
float="inline"
edges="bottom"
coords="none"
visible="area(b1,a2,b2)"
contents="R b1"/><hexboard size="3x3"
float="inline"
edges="bottom"
coords="none"
visible="area(c1,a3,c3)"
contents="R c1 a3"/><hexboard size="4x4"
float="inline"
edges="bottom"
coords="none"
visible="area(d1,a4,d4)"
contents="R d1 b3"/><hexboard size="5x5"
float="inline"
edges="bottom"
coords="none"
visible="area(e1,a5,e5)"
contents="R e1 c3 a5"/><hexboard size="6x6"
float="inline"
edges="bottom"
coords="none"
visible="area(f1,a6,f6)"
contents="R f1 d3 b5"/>

'''Answer:''' No. The first one in the sequence that is not connected is the one of height 8.

In fact, using a variant of [[Tom's move]], it is easy to see that even the following triangle, which has more red stones, is not an edge template:
<hexboard size="8x8"
float="inline"
edges="bottom"
coords="none"
visible="area(h1,a8,h8)"
contents="R h1 f3 d5 b7 c7,a8--e8"/>

To see why, imagine that the right edge is a blue edge and that all cells outside the carrier are occupied by Blue. Note that Blue gets a 2nd-and-4th row [[parallel ladder]]. Blue wins by playing the [[Tom's_move#Tall_variant|tall variant of Tom's move]]:
<hexboard size="8x8"
float="inline"
edges="bottom right"
coords="none"
visible="area(h1,a8,h8) g1--a7"
contents="R h1 f3 d5 b7 c7,a8--e8 B g1--a7 B 1:g2 R 2:h2 B 3:f5 R 4:g4 B 5:g3 R 6:h3 B 7:d6 R 8:g5 B 9:f7"/>

There is in fact a template of height 8 continuing the above sequence, but it requires slightly more space:

<hexboard size="8x9"
float="inline"
edges="bottom"
coords="none"
visible="area(h1,a8,i8,i6)"
contents="R h1 f3 d5 b7"/>
The corresponding template of height 9 requires this much space:

<hexboard size="9x11"
float="inline"
edges="bottom"
coords="none"
visible="area(i1,a9,k9,k7,i5)"
contents="R i1 g3 e5 c7 a9"/>

=== Seventh row template problem ===

Does there exist an [[edge template]] which guarantees a secure [[connection]] for a [[piece]] on the seventh row?

'''Answer:''' Yes. See [[Seventh row edge templates]].

=== Failure for monotone set-coloring games ===

I give two examples. The first is simple and highly-symmetric. The second goes 3-of-3: There are exactly 3 legal moves outside the set, and all 3 of them win.

simple and highly-symmetric:

Take a hexagonal [https://en.wikipedia.org/wiki/Bipyramid bipyramid], and give each equatorial edge an arrow pointing to a pole, such that the directions of the arrows alternate. Consider the [https://en.wikipedia.org/wiki/Maker-Breaker_game Maker-Breaker game] on the vertices of the resulting object, where the winning sets are the vertices of faces such that the equatorial edge's arrow points to the face's polar vertex.

Breaker is in a winning position: No matter what Maker's first move is, Breaker's first move is a pole. (If Maker's was not a pole, then there is symmetry between the poles until Breaker chooses one.) If neither of Maker's first two moves was a pole, then Breaker's second move is the other pole, winning for Breaker. Otherwise, Breaker pretends Maker's pole move was Maker's first move, and wins by [[Pairing_strategy|pairing]] using the equatorial edges whose arrows point to the pole Maker played.

Now, assume Maker plays a pole X. The analogue to the set A, is the set of vertices that are members of any minimal winning set which uses the vertex X. These are exactly the equatorial vertices, so in particular this set is non-empty.

However, if Breaker responds on the equator, then Breaker loses: Let Y be where Breaker just played. Maker responds on the equator, either opposite of Y, or the cell 120 degrees from Y whose immediate threat is adjacent to - rather than opposite from - Y. Breaker must defend against that threat, after which Maker plays the other equatorial vertex adjacent to where Maker just played. Lastly, Maker wins by playing either the pole Maker didn't already play, or continuing in the same direction on the equator.

going three-of-three:

This one is a [https://en.wikipedia.org/wiki/Maker-Breaker_game Maker-Breaker game] whose underlying set is {0,1L,1R,2L,2R,3L,3R,4}. There are exactly 6 minimal winning sets, and they are {0,1L,2L},{2L,3L,3R},{2L,3L,4} and the three formed by interchanging L with R.

Breaker is in a winning position: By symmetry, assume Maker does not play a R element. 0 and 2L each [[Dominated_cell#Switch-domination|switch-dominate]] 1L, so this leaves 0,2L,3L,4 as candidates for Maker's first move. If Maker's first move is 3L or 4, then Breaker can play 2L and win with the [[Pairing_strategy|pairing]] {0,1R},{2R,3R}. If Maker's first move is 2L or 0, then Breaker wins by playing any of 3L,3R,4 and using the [[Pairing_strategy|pairs]] {0,1L} and {1R,2R} and whichever of {3L,3R,4} Breaker hasn't yet played.

Now, assume Maker plays 0. The analogue to the set A, is the set of vertices that are members of any minimal winning set which uses the element 0. This is exactly {1L,1R,2L,2R}.

If Breaker responds in {1L,1R,2L,2R}, then Breaker loses: By symmetry, assume Breaker plays 1L or 2L. Maker responds at 2R, threatening to win immediately with 1R. Breaker must defend against that threat, after which Maker plays 3R, and wins with 3L or 4.

(As noted in the "Breaker is in a winning position" part here, all three of Breaker's moves outside of the analogue-of-A win for Breaker.)

[[category: Open problems]]
[[category: Forums]]

Theorems about templates

2024-04-21T05:51:35Z

Demer: fixed typo

There are a number of theorems about templates, some of which can be useful in play. Some of these theorems concern how to construct new templates from existing ones. Others concern how to play when templates overlap. Others explain why templates tend to have particular shapes.

== New templates from old ==

When we have theorems that allow us to construct new templates from known ones, there are fewer templates to memorize.

=== Corner clipping ===

We begin by observing that the following two positions are strategically equivalent:
A: <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(d1,a4,c4,d3)"
contents="B d1--d3 E x:c2 y:b3 z:c3"
/> B: <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(d1,a4,c4,d3)"
contents="B d1--d3 R c2 B c3 E w:b3"
/>
Indeed, if Red plays first in the region, then ''x'' [[captured cell|captures]] the entire triangle (''x'',''y'',''z'') in A, and ''w'' captures the corresponding triangle in B. Therefore, under [[optimal play]], Red achieves exactly the same thing in A as in B. Similarly, if Blue plays first in the region, ''y'' captures the whole triangle in A and ''w'' [[dead cell|kills]] the red stone and therefore captures the whole triangle in B. Therefore, under optimal play, Blue achieves the same thing in A as in B. It follows that A and B are equivalent.

Then we have the following theorem about edge templates:

'''Theorem 1 (corner clipping).''' Suppose an edge template has a corner of the form
A: <hexboard size="3x3"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="S blue:c1--c3"
/>
Here, the blue-shaded cells must not be part of the template, i.e., they must be outside of its [[carrier]]. Then the pattern where this corner has been replaced by
B: <hexboard size="3x3"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="S blue:c1--c3--b3 R b2"
/>
is also an edge template. The converse is also true, i.e., if some template has a corner of shape B, the corresponding pattern with shape A is also a template.

'''Proof.''' Since the shaded hexes are outside the carrier, they may as well be blue stones. Then by the above observation, A and B are completely equivalent, so if some pattern containing A is connected, then so is the corresponding pattern containing B. Moreover, it is easy to see that if removing one cell from the carrier of A would yield a connected shape, then the same could be achieved by removing one cell or the red stone from B, and vice versa. Therefore, the template containing A is minimal if and only if the template containing B is minimal. □

'''Examples'''

Corner clipping shows that the ziggurat is equivalent to edge templates [[Edge template III2b|III2b]] and [[Edge template III2g|III2g]], as well as a very compact 3-stone template:

<hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="R c1"
/> ⇔ <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1 d3"
contents="R c1 d2"
/> ⇔ <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1 a3"
contents="R c1 b2"
/> ⇔ <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1 a3 d3"
contents="R c1 b2 d2"
/>

[[Edge template IV2a]]:

<hexboard size="4x4"
float="inline"
coords="none"
edges="bottom"
visible="-a1--a3 b1"
contents="R c1 d1"
/> ⇔ <hexboard size="4x4"
float="inline"
coords="none"
edges="bottom"
visible="-a1--a3 b1 d4"
contents="R c1 d1 d3"
/>

'''Remark.''' For the clipped corner theorem to hold, the corner must be of this shape
<hexboard size="3x3"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="S blue:c1--c3"
/>
and not merely that one:
<hexboard size="3x3"
edges="bottom"
coords="none"
visible="-a1 a2 b1 c1"
contents="S blue:c1--c3"
/>
In other words, the cell on the 3rd row should not be part of the carrier. However, if the corner is merely of the latter form, the clipped template is still valid. It may not be minimal. For example, consider [[edge template IV2b]]:
<hexboard size="4x5"
coords="none"
edges="bottom"
visible="-a1--a3 b1 e1"
contents="R c1 b2"
/>
Clipping the right corner is no problem. If we clip the left corner, the resulting pattern is connected, but not minimal:
<hexboard size="4x5"
coords="none"
edges="bottom"
visible="-a1--a3 b1 e1 a4"
contents="R c1 b2 b3 E *:(e2--e4)"
/>
The three hexes marked "*" could be removed from the carrier while still remaining connected.

=== Large corner clipping ===

Observe that the following positions are equivalent:
A: <hexboard size="4x2"
float="inline"
edges="none"
coords="none"
visible="-b4"
contents="B a1--b1--b3--a4 R a3"
/> B: <hexboard size="4x2"
float="inline"
edges="none"
coords="none"
visible="-b4"
contents="B a1--b1--b3--a4 R a2"
/>
Indeed, if Red moves in the region, A and B become identical. If Blue moves in the region, Blue [[dead cell|kills]] Red's stone, so again A and B also become identical.

Then we have the following theorem about edge templates:

'''Theorem 2 (large corner clipping).''' Suppose an edge template has a corner of the form
A: <hexboard size="4x4"
edges="bottom"
coords="none"
visible="area(b4,d4,d1,c1)"
contents="S blue:c1--d1--d4"
/>
Here, the blue-shaded cells must not be part of the template, i.e., they must be outside of its [[carrier]]. Then the pattern where this corner has been replaced by
B: <hexboard size="4x4"
edges="bottom"
coords="none"
visible="area(b4,d4,d1,c1)"
contents="S blue:c1--d1--d4--c4 R c2"
/>
is also an edge template. The converse is also true, i.e., if some template has a corner of shape B, the corresponding pattern with shape A is also a template.

'''Proof.''' By ordinary corner clipping, A is equivalent to
<hexboard size="4x4"
edges="bottom"
coords="none"
visible="area(b4,d4,d1,c1)"
contents="S blue:c1--d1--d4--c4 R c3"
/>
which is in turns equivalent to B by the above observation. □

'''Examples'''

Large corner clipping shows that the ziggurat is equivalent to [[edge template III2a]]:

<hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="R c1"
/> ⇔ <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1 d3"
contents="R c1 d1"
/>

Similarly, we can construct several new templates from [[edge template IV2a]]:
<hexboard size="4x7"
float="inline"
coords="none"
edges="bottom"
visible="area(e1,c2,a4,g4,g2,f1)"
contents="R e1"
/> ⇔ <hexboard size="4x7"
float="inline"
coords="none"
edges="bottom"
visible="area(e1,c2,a4,g4,g2,f1)-a4"
contents="R arrow(12):e1 c2"
/> ⇔ <hexboard size="4x7"
float="inline"
coords="none"
edges="bottom"
visible="area(e1,c2,a4,g4,g2,f1)-g4"
contents="R arrow(12):e1 g2"
/>
There are of course additional possibilities, such as clipping both corners, combining large and ordinary corner clipping, etc.

=== Corner bending ===

The idea of corner bending is similar to that of corner clipping. We again start with an observation about two positions. This time, we claim that B is at least as good for Red as A.
A: <hexboard size="2x2"
float="inline"
edges="bottom"
coords="none"
visible="-a1"
contents="B b1 b2 E x:a2"
/> B: <hexboard size="2x2"
float="inline"
edges="bottom"
coords="none"
visible="-a1"
contents="B a2 R b2 E y:b1"
/>
Indeed, if Red plays first in B, then ''y'' [[dead cell|kills]] the blue stone and therefore [[captured cell|captures]] the whole region, which is certainly at least as good as anything that Red could achieve in A. On the other hand, if Blue plays first in A, then Blue occupies the whole region, which is certainly at least as bad for Red as anything Blue could achieve in B. Therefore, if any position containing A is winning for Red, then so is the corresponding position containing B.

We obtain the following theorem:

'''Theorem 3 (corner bending).''' Suppose an edge template has a corner of the form
A: <hexboard size="2x2"
float="inline"
edges="bottom"
coords="none"
visible="-a1"
contents="S blue:(b1 b2) E x:a2"
/>
Here, the blue-shaded cell must not be part of the template, i.e., it must be outside of its [[carrier]]. Then the pattern where this corner has been replaced by
B: <hexboard size="2x2"
float="inline"
edges="bottom"
coords="none"
visible="-a1"
contents="S blue:a2 R b2 E y:b1"
/>
is still connected. (It may fail to be an edge template only because it may fail to be minimal).

'''Proof.''' Since the shaded hexes are outside the carrier, they may as well be blue stones. Then by the above observation, B is at least as good for Red as A, so if some region containing A is connected for Red, then so is the same region containing B. □

'''Examples'''

The following is a [[ziggurat]], followed by ziggurats with one or two bent corners.

<hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="R c1"
/> <hexboard size="3x5"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1 d3 e1 d4"
contents="R c1 e3"
/> <hexboard size="3x5"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1 c1 b3"
contents="R d1 a3"
/> <hexboard size="3x6"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1 c1 b3 f1 e3"
contents="R d1 a3 f3"
/>

=== Crescenting ===

We begin by observing that the following three positions are strategically equivalent:

A: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(b1,a3,c4,d3,d1)"
contents="B all R a3 z:b3 B x:b2 y:c2 w:c3"
/> B: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(b1,a3,c4,d3,d1)"
contents="B all R a3 z:b3 x:b2 y:c2 w:c3"
/> C: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(b1,a3,c4,d3,d1)"
contents="B all R a3 x:b2 y:c2 E z:b3 w:c3"
/>

Indeed, A and B are equivalent because x, y, and w are [[dead cell|dead]], and B and C are equivalent because z and w are [[captured cell|captured]]. We then have the following theorem about templates:

'''Theorem 4 (crescenting).''' Suppose some (edge or interior) template has a piece of boundary of the form
<hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(b1,a3,c4,d3,d1)"
contents="S blue:all none:a3,b3 R a3 b3"
/>
Here, as usual, the blue-shaded cells are not part of the template. Then the pattern where this area has been replaced by
<hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(b1,a3,c4,d3,d1)"
contents="S blue:all none:a3,b3,b2,c2,c3 R a3 b2 c2"
/>
is also a template. Moreover, the converse also holds. (Caveat: the construction preserves minimality with respect to empty cells in the carrier. In some boundary cases, it is possible that some of the red stones are not actually necessary in one or the other template).

'''Proof.''' Since the shaded hexes are outside the carrier, they may as well be blue stones. Then the equivalence follows by the above observation. □

'''Examples'''

Many templates with two adjacent stones have crescented versions. Indeed, the crescent itself is such a template:
<hexboard size="3x2"
float="inline"
edges="none"
coords="none"
visible="-b3"
contents="R a1 a3 b1"
/> ⇔ <hexboard size="4x3"
float="inline"
edges="none"
coords="none"
visible="area(a2,a4,c2,c1,b1)"
contents="R a2 a4 b1 c1"
/>
Here is a crescented version of [[edge template IV2a]]. Note that crescenting can also be applied recursively, resulting in templates that resemble a [[Interior_template#The_long_crescent|long crescent]].
<hexboard size="4x4"
float="inline"
coords="none"
edges="bottom"
visible="area(c1,b2,a4,d4,d1)"
contents="R c1 d1"
/> ⇔ <hexboard size="5x5"
float="inline"
coords="none"
edges="bottom"
visible="area(d1,b3,a5,d5,d3,e2,e1)"
contents="R c2 d1 e1"
/> ⇔ <hexboard size="6x6"
float="inline"
coords="none"
edges="bottom"
visible="area(e1,b4,a6,d6,d4,f2,f1)"
contents="R c3 d2 e1 f1"
/>

Often, the crescent-like shape can be more generally replaced with any capped [[flank]]. See also [[Flank#Edge_templates_from_capped_flanks|edge templates from capped flanks]].

=== Alternative connection up ===

Some templates, such as [[Tom's move]], have an "alternative connection up". There is a general theorem about this. We begin by observing that from Red's point of view, C is at least as good as B, and B is at least as good as A:

A: <hexboard size="3x4"
float="inline"
edges="none"
coords="none"
visible="area(a2,b3,d1,c1)"
contents="R c1 b2 b3 B a2 d1 E c2"
/> B: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a2,b3,d1,c1)"
contents="R c1 b3 B a2 R b2 B w:c2 E z:d1"
/> C: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a2,b3,d1,c1)"
contents="R c1 b3 B a2 E x:b2 y:c2 z:d1"
/>

To see that B is at least as good for Red as A, note that if Red plays first in the region, then Red plays z in B, which [[dead cell|kills]] w and is at least as good as anything Red could do in A. If Blue plays first in the region, A and B become identical. To see that C is at least as good for Red as B, note that Red can get at least one of x and y by defending the [[bridge]]. No matter which way the bridge goes, the result is identical or better for Red than B.

'''Theorem 5 (alternative connection up).''' Suppose some (edge or interior) template has a piece of boundary of the form

A: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a3,b4,d2,d1,c1)"
contents="S blue:all none:b4,c3 R arrow(12):b4"
/>

Here, as usual, the blue-shaded cells are not part of the template, but the stone marked "↑" must of course be connected up. Then the patterns where this area has been replaced by

B: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a3,b4,d2,d1,c1)"
contents="S blue:all none:b4,c3,b3,d2,c2 R b4 arrow(12):c2 R b3 B c3"
/> or C: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a3,b4,d2,d1,c1)"
contents="S blue:all none:b4,c3,b3,d2,c2 R b4 arrow(12):c2"
/>

are also connected. (They may fail to be templates only because they may fail to be minimal).

'''Proof.''' Consider the template containing A. Since the shaded hexes are outside the carrier, they may as well be blue stones, except that we need to connect the stone marked "↑" to something, which we can without loss of generality do like this:
<hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a3,b4,d2,d1,c1)"
contents="B a3 b2 d1 d2 R a:b4 b:b3 c:c2 arrow(12):c1"
/>
The fact that the stones "b" and "c" are adjacent to cells in the template does not matter, because "a" is adjacent to the same cells anyway. By the above observation, each of the following is at least as good for Red, and therefore also connected:
<hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a3,b4,d2,d1,c1)"
contents="B a3 b2 d1 R b4 c2 arrow(12):c1 R b3 B c3"
/> and <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a3,b4,d2,d1,c1)"
contents="B a3 b2 d1 R b4 c2 arrow(12):c1"
/>
Therefore, the patterns using B and C are connected, proving the theorem. □

'''Example'''

[[Tom's move]] ensures that the following is connected:
<hexboard size="5x7"
edges="bottom"
coords="none"
visible="-(a2 a1 b1 f1 g1 g2)"
contents="R arrow(12):b2 a3 a4 B b3"
/>
By Theorem 5, the following are therefore also connected:
<hexboard size="6x8"
float="inline"
edges="bottom"
coords="none"
visible="-g2 h2 h3 f1--h1 -area(a1,a6,c1)"
contents="R b4 b5 c3 B a6 c4 R arrow(12):d1 R c2 B d2"
/> and <hexboard size="6x8"
float="inline"
edges="bottom"
coords="none"
visible="-g2 h2 h3 f1--h1 -area(a1,a6,c1)"
contents="R b4 b5 c3 B a6 c4 R arrow(12):d1"
/>
This is exactly the "[[Tom's_move#Alternative_connection_up|alternative connection up]]" of Tom's move.

== Ladder creation templates from templates ==

By an argument similar to the first observation above, we observe that the following three positions are strategically equivalent:
A: <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(c1,a3,d3,d1)"
contents="B c1--c3,d1--d3 E x:b2 y:a3 z:b3"
/> B: <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(c1,a3,d3,d1)"
contents="B c1--c3,d1--d3 R b2 B a3 E w:b3"
/> C: <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(c1,a3,d3,d1)"
contents="B c1,b3,d1--d3 R c2 E x:b2 y:a3 z:c3"
/>
Indeed, whoever plays first in each region captures the entire region: Red by playing at ''x'' or ''w'', and Blue by playing at ''y'' or ''w''.

An interesting application of this is getting a 2nd row [[ladder creation template]] from an ordinary template.

'''Theorem 6 (ladder creation template from template).''' Suppose an edge template has a corner of the form
A: <hexboard size="3x4"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="S blue:c1--c3,d1--d3"
/>
where as usual, the blue-shaded cells are not part of the template. Then the pattern where this corner has been replaced by either one of
B: <hexboard size="3x4"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="S blue:c1--c3,d1--d3 B a3 E arrow(3):b2,b3"
/>
C: <hexboard size="3x4"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="S blue:c1,d1--d3 B b3 E arrow(3):c2,c3"
/>
is a 2nd row ladder creation template. The converse is also true, i.e., if some pattern with a corner of shape B or C is a 2nd row ladder creation template, then the corresponding pattern with shape A is an edge template.

'''Proof.''' For the equivalence of A and B, note that by the above observation, A is connected if and only if B' is connected:
B': <hexboard size="3x4"
edges="bottom"
coords="none"
visible="-a1 a2 b1"
contents="S blue:c1--c3,d1--d3 B a3 R b2 E b3"
/>
By essentially Theorem 1 of the article on the [[theory of ladder escapes]], B' is connected if and only if B creates a 2nd row ladder. Finally, since it is an "if and only if", it follows that if A is minimal, so is B, and vice versa. The argument for the equivalence of A and C is analogous. □

'''Example'''

From the [[ziggurat]], we get the following ladder creation templates:
<hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(c1,a3,d3,d1)"
contents="R c1 B c3 E arrow(3):d2,d3"
/> <hexboard size="3x5"
float="inline"
edges="bottom"
coords="none"
visible="area(c1,a3,e3,e2,d1)"
contents="R c1 B d3 E arrow(3):e2,e3"
/> <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(c1,a3,d3,d1)"
contents="R c1 B b3 E arrow(9):b2,a3"
/> <hexboard size="3x5"
float="inline"
edges="bottom"
coords="none"
visible="area(d1,b2,a3,e3,e1)"
contents="R d1 B b3 E arrow(9):b2,a3"
/>

What we learn from this is that a Blue [[intrusion]] into the very corner of Red's template, or the cell right next to the corner, is not usually a good idea. Red can reconnect by creating a 2nd row ladder escape, potentially far away. This will often allow Red to play a [[minimaxing]] response. In particular, if Red already has a 2nd row ladder escape, the intrusion is not even valid (it does not even threaten to disconnect Red).

It is worth remarking that if the edge template's corner is merely of this shape,
<hexboard size="3x4"
edges="bottom"
coords="none"
visible="-a1 a2 b1 c1 d1"
contents="S blue:c1--c3,d1--d3"
/>
then the left-to-right direction of Theorem 6 is still valid: Given an edge template, we still obtain a ladder creation template in two different ways. However, in this case, the latter may not be minimal.

== Overlapping templates ==

When templates [[Template#Overlapping_templates|overlap]], they are usually not both valid. However, there are some exceptions where templates can overlap and still be valid. It is useful to know them.

=== Edge template II in the overlap ===

'''Theorem 7.''' If the region in which two edge templates overlap is [[edge template II]], then both templates remain valid.
<hexboard size="3x4"
coords="hide"
edges="bottom"
visible="b3 c2 c3"
contents="R c2"
/>

'''Proof.''' The two empty hexes in edge template II are [[captured cell|captured]], and therefore they can be replaced by red stones without changing the strategic value.
<hexboard size="3x4"
coords="hide"
edges="bottom"
visible="b3 c2 c3"
contents="R c2 b3 c3"
/>
Overlapping templates are only invalid if there are empty cells in the overlap. □

'''Example'''

<hexboard size="4x4"
float="inline"
coords="hide"
edges="bottom"
visible="area(d1,a4,d4)"
contents="R d1 d3"
/> + <hexboard size="4x4"
float="inline"
coords="hide"
edges="bottom"
visible="area(d1,a4,d4)"
contents="R d1 b3"
/> = <hexboard size="4x6"
float="inline"
coords="hide"
edges="bottom"
visible="area(d1,a4,f4,f1) - e1"
contents="R d1 d3 f1"
/>
Both templates remain valid, i.e., all three red stones can be connected to the edge simultaneously.

=== The ziggurat theorem ===

The following theorem is due to Eric Demer.

'''Theorem 8 (ziggurat theorem).''' Consider a [[ziggurat]].
<hexboard size="3x4"
visible="area(a3,d3,d1,c1)"
edges="bottom"
coords="none"
contents="R 1:c1 E x:a3 y:d3 z:c2"
/>
If the ziggurat overlaps another edge template in the cell ''x'', and/or overlaps another edge template in the cell ''y'', all templates (i.e., the ziggurat itself and its neighboring templates) remain valid. If Blue plays in the overlap at ''x'' or ''y'', Red can restore all templates by playing at ''z''. Moreover, this even remains true if the ziggurat has not been completed yet (i.e., if the template stone 1 has not yet been played).

'''Proof.''' If Blue plays at ''x'' or ''y'' (or both), clearly Red playing at ''z'' defends the ziggurat. What we must show is that it defends the neighboring templates as well. But if Red plays at ''z'', then the two hexes just below ''z'' are [[captured cell|captured]].
<hexboard size="3x4"
visible="area(a3,d3,d1,c1)"
edges="bottom"
coords="none"
contents="R 1:c1 B x:a3 y:d3 R z:c2 R b3 c3"
/>
Then the neighboring templates are valid by Theorem 3 above (the corner bending theorem).

The other thing we must show is that if Blue starts by playing in the ziggurat anywhere other than at ''x'' or ''y'', then Red can always reconnect in a way that either captures or no longer needs ''x'' and ''y''. Indeed, if Blue plays anywhere on the right, Red can play like this:
<hexboard size="3x4"
visible="area(a3,d3,d1,c1)"
edges="bottom"
coords="none"
contents="R 1:c1 E x:a3 y:d3 B d1 c2 d2 c3 R b2"
/>
which captures ''x'' and no longer needs ''y''. And if Blue plays on the left, Red responds like this, which captures ''y'' and no longer needs ''x'':
<hexboard size="3x4"
visible="area(a3,d3,d1,c1)"
edges="bottom"
coords="none"
contents="R 1:c1 E x:a3 y:d3 B b2 b3 R d2"
/>
□

'''Example'''

The classic application of the ziggurat theorem is [[edge template IV2c]]:
<hexboard size="4x7"
coords="hide"
edges="bottom"
visible="-area(a1,a3,d1) -g1"
contents="R e1 f1 B e2 E a:d2 b:f2 c:d4"
/>
Red is threatening to play at ''a'' or ''b'', getting a ziggurat each way. Blue's only hope is to play in the overlap. Alas, by the ziggurat theorem, this does not work. Red knows that she should play at 2 (the symmetric move would of course also have worked):
<hexboard size="4x7"
coords="hide"
edges="bottom"
visible="-area(a1,a3,d1) -g1"
contents="R e1 f1 B e2 B 1:d4 R 2:c3"
/>
Now Blue is sure to lose:
<hexboard size="4x7"
coords="hide"
edges="bottom"
visible="-area(a1,a3,d1) -g1"
contents="R e1 f1 B e2 B d4 R c3 B 3:d2 R 4:f2 B 5:f3 R 6:e3 B 7:e4 R 8:d3"
/>
There are other ways for Red to connect here; for one, Red's moves 2, 4, 6 could have been played in a different order. But by using the ziggurat theorem, Red can easily know what to do without having to think hard, and can concentrate on other trickier areas of the board.

'''Generalizations.''' There are many other edge templates (besides the ziggurat) for which a version of the ziggurat theorem holds, but it is not known whether it holds for all templates of the appropriate shape. Perhaps a list of such templates could be added here at some point.

== The shape of templates ==

You may have noticed that many edge templates resemble each other: their boundaries tend to follow a relatively small number of possible shapes. This is not a coincidence. Some of it is due to the following theorems.

Observation: the two empty cells in the following pattern are [[captured cell|captured]] by Blue.
<hexboard size="4x3"
edges="none"
coords="none"
visible="-a1--a3 b4 c4"
contents="R a4 B b1 c1--c3"
/>
Indeed, if Red plays in one of these cells, Blue can play in the other, [[dead cell|killing]] Red's stone. From this, we immediately get the following theorem:

'''Theorem 9.''' There is no edge template with an empty corner of height 2, i.e., with a corner of this shape:
<hexboard size="3x2"
edges="bottom"
coords="none"
contents="S blue:(a1 b1--b3)"
/>
Here, as in previous theorems, the blue-shaded cells indicate the outside of the template, i.e., they are not part of the carrier.

'''Proof.''' Since the shaded hexes are outside the carrier, they may as well be blue stones. But then the two empty cells are captured by the previous observation, which means they can also be filled in with blue stones, contradicting the minimality of the template. □

We note that the theorem only says that the corner of an edge template cannot be of height 2 if that corner is empty. When there are stones in the corner, height 2 is possible, as in the following examples:
<hexboard size="2x2"
float="inline"
edges="bottom"
coords="none"
visible="-a1"
contents="R b1"
/> <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="-a1 a2 b1 d1"
contents="R arrow(12):c1 d3"
/>

Observation: The following two regions are equivalent:
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
contents="B a1--c1--c3 E x:a3"
/> <hexboard size="3x3"
float="inline"
coords="none"
edges="none"
contents="B a1--c1--c3 y:b2 E x:a3 "
/>
To see why, first assume that Blue plays first in the region. Then Blue x captures the entire region, so the stone at y no longer matters. Next, assume that Red plays first in the region. If Red plays anywhere other than x, then Blue x [[dead cell|kills]] the red stone. If Red plays at x, then y is [[dead cell|dead]], so the stone at y no longer matters. As a consequence, we get the following theorem:

'''Theorem 10.''' There is no edge template with a piece of boundary of this shape:
<hexboard size="3x3"
float="inline"
coords="none"
edges="none"
contents="S blue:a1--c1--c3 E y:b2"
/>
Here, as in previous theorems, the blue-shaded cells indicate the outside of the template, i.e., they are not part of the carrier.

'''Proof.''' Since the shaded hexes are outside the carrier, they may as well be blue stones. Then by the previous observation, y might as well be blue, contradicting the minimality of the template. □

Once again, we remark that Theorem 10 presupposes that there are no stones in the relevant portion of the template. For example, the [[ziggurat]] is an edge template that ends in two columns of height 3, but this does not contradict Theorem 10 due to the presence of a red stone in one of these columns.

'''Theorem 11.''' There is no edge template with a corner of this shape:
<hexboard size="4x3"
float="inline"
coords="none"
edges="bottom"
visible="-a1 a2 b1 c4"
contents="S blue:(c1--c3 b4) E y:b3"
/>
As always, the blue-shaded cells indicate the outside of the template, i.e., they are not part of the carrier.

'''Proof.''' This is very similar to Theorem 10. By the previous observation, y might as well be blue, contradicting the minimality of the template. □

Theorems 7–9 explain why the rightmost few columns of edge templates frequently have one of these shapes
<hexboard size="5x3"
float="inline"
coords="none"
edges="bottom"
visible="area(a1,a5,c5,c3)"
/> <hexboard size="5x2"
float="inline"
coords="none"
edges="bottom"
visible="area(a1,a5,b5,b3)"
/>
and never one of these, unless the template contains stones in those regions:
<hexboard size="5x3"
float="inline"
coords="none"
edges="bottom"
visible="area(a1,a5,c5,c3)-b2"
/> <hexboard size="5x3"
float="inline"
coords="none"
edges="bottom"
visible="area(a1,a5,c5,c3)+b1"
/> <hexboard size="5x3"
float="inline"
coords="none"
edges="bottom"
visible="area(a2,a5,c5,c4)"
/> <hexboard size="4x2"
float="inline"
coords="none"
edges="bottom"
visible="-b4"
/>

[[Category: Theory]]

User:Hexanna

2024-04-19T19:32:25Z

Demer: replied

[[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.

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

* katahex_model_20220618.bin.gz (I'll call this the "strong" net) appears significantly stronger than the "default" net.
* Swap map for 19×19 generated with the strong net, 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%).
** Key takeaways: The swap map at [[Swap rule#Size 19]] uses [https://pic4.zhimg.com/v2-7287c3a2a4e948da89c3ccad38cea82f_r.jpg data] that is almost certainly from the "weak" net. Compared to the weak net, the strong net notably thinks a19, n3—p3, and k4—l4 are stronger. I personally trust the strong net's evaluations more; I think it's dubious that the weak net thought l4 was a very fair opening. The nets disagree on whether e3 is winning or losing, though it's so close to 50% that the difference isn't meaningful.

<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)
72:(i4 k16)
67:(j4 j16)
81:(k4 i16)
94:(l4 h16)
69:(q2 c18)
96:(b17 r3)
77:(a2 s18)
67:(b2 r18)
56:(c2 q18)
100:(a3 s17)
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)
81:(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.

==Article ideas==

* '''Motifs''' — very loosely related to joseki; small local patterns that occur in the middle of the board, usually representing optimal play from at least one side but not necessarily both sides
** Motifs have some notion of '''"local efficiency"''' (not to be confused with [[efficiency]]) — some motifs are, on average, good or bad for a particular player. Strong players anecdotally try to play locally efficient moves on large boards where calculating everything is impractical. It would be useful to have some of these rules of thumb written down. Can be thought of as a generalization of dead/captured cells, where LE(dead cell) = 0, and LE(X) ≤ LE(Y) if Y capture-dominates X.
** Here are some examples. In the first motif, Red 1 is often a weak move. Blue's best response is usually at a, or sometimes at b or c as part of a minimaxing play. But d is rarely (possibly never) the best move, because Red can respond with a, and Blue's central stone is now a dead stone. So, for any reasonable working definition of "local efficiency" LE, we have LE(d) < LE(a), and LE(b) = LE(c) due to symmetry. KataHex suggests that LE(b) < LE(a).

<hexboard size="5x5"
coords="none"
edges="none"
contents="R b3 B c3 R 1:d2 E a:c2 b:b4 c:d3 d:c4"
/>

Sometimes, a player will attempt to minimax by placing two stones adjacent to each other, like the unmarked blue stones below. (This is a common human mistake on 19×19; adjacent stones are typically less locally efficient than stones a bridge apart.) Red has several options, such as the adjacent block (*), though a far block is often possible too. It would be enlightening to know, absent other considerations, which block is the most "efficient" for Red, so that on a large board, Red could play this block without thinking too hard. Of course, in general the best move depends on the other stones on the board, and there's no move that strictly dominates another. The best move may even plausibly be to "[[tenuki|play elsewhere]]." Provisionally, KataHex thinks playing at one of A, or the far block at B, is a better first move for Red.

<hexboard size="4x4"
coords="none"
edges="none"
contents="B b1 c1 E *:c2 A:(a1 d1) B:a4"
/>

----

[[User:Fjan2ej57w|Fjan2ej57w]]'s question 7, "how much space of an empty board would be filled if both sides play optimally":

[https://mathoverflow.net/questions/302821/length-of-optimal-play-in-hex-as-a-function-of-size Stack Overflow answer] for reference. My conjecture is that Hex without swap asymptotically requires Θ(n^2) cells, and more generally, a Demer handicap of Θ(f(n)) stones requires Θ(n^2/f(n)) cells, for all f(n) between Θ(1) and Θ(n). My intuition is that on 1000000×1000000 Hex, the first-player advantage is minuscule, and even a handicap of n^(1/2) = 1000 stones, say spaced out evenly across the short diagonal, would require on the order of "1000 columns and 1000000 rows", n^(3/2), to convert to a final connection. Another interesting question is to find a constructive winning strategy with an o(n) (sub-linear) handicap.

reply by [[User:Demer|Demer]]: Even one with n/6 - ω(1) handicap would be interesting. (improving on https://webdocs.cs.ualberta.ca/~hayward/papers/handicap.pdf )

Solutions to puzzles

2024-02-12T06:38:50Z

Demer: /* Puzzle 5 */ fixed wrong cells

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, bridging to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in [[Solutions to puzzles#Puzzle 2_2|puzzle 2]].

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]]:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with b2. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the d1,b2 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red.

Showing that b4 wins for Red:

<hexboard size="5x5"
contents="R e2 a1 1:b4 B b5 e1"
/>

If Blue plays to the upper-left of the [[short diagonal]] ''other than'' in {c2,b3}, then Red responds with c2 and defends the bridge. c2 connects to the top via d1 or c1 or b2, and connects to the bottom with a very short [[switchback]].

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:c2 5:b3 7:d4 9:d2 B b5 e1 2:a3 4:c3 6:a5 8:c4"
/>

If Blue plays to the lower-right of the [[short diagonal]] ''other than'' at d4, then Red plays a5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. Thus Blue plays b3 instead. Red responds with d2, and a1 escapes that ladder.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:a5 B b5 e1 S blue:(d3 e3 e4 e5 d5 c5 c4)"
/>

Blue must continue with c3, and what's left of this case is checking that Red can connect e2 down no matter which of the above shaded cells was Blue's first move. If that move was neither d3 nor e3, then Red bridges to d4, and connects down with c4 or c5 or d5. Otherwise, Red plays whichever of d3,e3 Blue didn't play. Since Red has a5 and b4, b5 is [[Dead cell|dead]], so Red ladders to b4.

If Blue plays d4, then Red plays c4, capturing a5 and c5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. If Blue plays b3, then Red plays d2, bridging from c4 and laddering to a1.

This leaves the following tries for Blue.

<hexboard size="5x5"
contents="R e2 a1 1:b4 B b5 e1 S d2 c2 c3 b3 a5"
/>

c2 loses to a3, using [[Edge template III2d]]

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:a3 5:d4 7:d2 B b5 e1 2:c2 4:a5 6:c4"
/>

and a [[switchback]], as shown above.

If Blue plays b3, then Red plays d2, and a1 escapes that ladder. Towards the bottom, if Blue plays outside of {c4,c3,d3}, then Red can play c4 and defend the bridge from d2, and if Blue plays outside of {d4,d5,c5}, then Red can play d4 and defend the c5+d5 [[Captured cell|capture]].

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d2 B b5 e1 2:b3"
/>

In the former case, Red connects down with a5 or c5 or e4. In the latter case, Red connects down with e3 or d3 or c3, since c3 threatens c4 and a5. The sets {c4,c3,d3} and {d4,d5,c5} are disjoint, so Red gets at least one of those, and thereby connects down for a win.

If Blue plays c3, then Red plays d4, capturing e2 and b4 to the bottom. Blue must respond in {d2,c2,b3}, since otherwise Red plays c2, connecting down with d2 or b3 and connecting to the top with d1 or c1 or b2. If Blue plays d2 or c2, then Red plays a3, winning with [[Edge template III2d]]. If Blue plays b3, then Red plays d2, and wins by laddering to a1.

If Blue plays d2, then Red plays c3, connecting down with a5 or d4. Towards the top, blue c2 loses to a3, forming [[Edge template III2d]], and Blue's other moves all lose to red c2, connecting with d1 or c1 or b2.

Blue's only remaining try is a5. Red responds to that with d4, [[Captured cell|capturing]] c5 and d5.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 c5 d5 B b5 e1 2:a5"
/>

Red's main threat is d4, laddering to a1.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 c5 d5 5:d2 B b5 e1 2:a5 S a2 b1 b2 c1 c2 d1 d2 d3 e3"
/>

If Blue plays in {a2,b1,b2,c1,d1}, then Red plays c2. This connects to the top with d1 or c1 or b2, and connects down with a [[Interior template#The box|box]].

If Blue plays c2, then Red plays d2. This connects down with a [[Interior template#The mouth or trapezoid|table]], and connects to the top with d1 or a3, since a3 forms [[Edge template III2d]].

If Blue plays d2, then Red plays c3. This has a bridge down, and connects to the top for the same reason as without the a5-b4 exchange.

The e2-d4 bridge is [[Bolstered template|bolstered]] at e3 ([[Equivalent patterns#Edge equivalence|by Blue's edge]]), so e3 is no better for Blue than d3.

Thus, the only remaining case is 4. d3 .

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 5:c4 c5 d5 B b5 e1 2:a5 4:d3"
/>

Against that, Red plays 5. c4 .

This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. Thus Blue plays b3 instead. Red responds with d2, connecting down via c3 or e3, and connecting to the top because a1 escapes the ladder.

Therefore 1. b4 does win for Red.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-02-12T06:31:26Z

Demer: /* Puzzle 5 */ changed phrasing to correspond to who's turn it is

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, bridging to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in [[Solutions to puzzles#Puzzle 2_2|puzzle 2]].

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]]:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with b2. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the d1,b2 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red.

Showing that b4 wins for Red:

<hexboard size="5x5"
contents="R e2 a1 1:b4 B b5 e1"
/>

If Blue plays to the upper-left of the [[short diagonal]] ''other than'' in {c2,b3}, then Red responds with c2 and defends the bridge. c2 connects to the top via d1 or c1 or b2, and connects to the bottom with a very short [[switchback]].

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:c2 5:b3 7:d4 9:d2 B b5 e1 2:a3 4:c3 6:a5 8:c4"
/>

If Blue plays to the lower-right of the [[short diagonal]] ''other than'' at d4, then Red plays a5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. Thus Blue plays b3 instead. Red responds with d2, and a1 escapes that ladder.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:a5 B b5 e1 S blue:(d3 e3 e4 e5 d5 c5 c4)"
/>

Blue must continue with c3, and what's left of this case is checking that Red can connect e2 down no matter which of the above shaded cells was Blue's first move. If that move was neither d3 nor e3, then Red bridges to d4, and connects down with c4 or c5 or d5. Otherwise, Red plays whichever of d3,e3 Blue didn't play. Since Red has a5 and b4, b5 is [[Dead cell|dead]], so Red ladders to b4.

If Blue plays d4, then Red plays c4, capturing a5 and c5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. If Blue plays b3, then Red plays d2, bridging from c4 and laddering to a1.

This leaves the following tries for Blue.

<hexboard size="5x5"
contents="R e2 a1 1:b4 B b5 e1 S d2 c2 c3 b3 a5"
/>

c2 loses to a3, using [[Edge template III2d]]

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:a3 5:d4 7:d2 B b5 e1 2:c2 4:a5 6:c4"
/>

and a [[switchback]], as shown above.

If Blue plays b3, then Red plays d2, and a1 escapes that ladder. Towards the bottom, if Blue plays outside of {c4,c3,d3}, then Red can play c4 and defend the bridge from d2, and if Blue plays outside of {d4,d5,c5}, then Red can play d4 and defend the c5+d5 [[Captured cell|capture]].

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d2 B b5 e1 2:b3"
/>

In the former case, Red connects down with a5 or c5 or e4. In the latter case, Red connects down with e3 or d3 or c3, since c3 threatens c4 and a5. The sets {c4,c3,d3} and {d4,d5,c5} are disjoint, so Red gets at least one of those, and thereby connects down for a win.

If Blue plays c3, then Red plays d4, capturing e2 and b4 to the bottom. Blue must respond in {d2,c2,b3}, since otherwise Red plays c2, connecting down with d2 or b3 and connecting to the top with d1 or c1 or b2. If Blue plays d2 or c2, then Red plays a3, winning with [[Edge template III2d]]. If Blue plays b3, then Red plays d2, and wins by laddering to a1.

If Blue plays d2, then Red plays c3, connecting down with a5 or d4. Towards the top, blue c2 loses to a3, forming [[Edge template III2d]], and Blue's other moves all lose to red c2, connecting with d1 or c1 or b2.

Blue's only remaining try is a5. Red responds to that with d4, [[Captured cell|capturing]] c5 and d5.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 c5 d5 B b5 e1 2:a5"
/>

Red's main threat is d4, laddering to a1.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 c5 d5 5:d2 B b5 e1 2:a5 S a2 b1 b2 c1 c2 d1 d2 d3 e3"
/>

If Blue plays in {a2,b1,b2,c1,d1}, then Red plays c2. This connects to the top with d1 or c1 or b2, and connects down with a [[Interior template#The box|box]].

If Blue plays c2, then Red plays d2. This connects down with a [[Interior template#The mouth or trapezoid|table]], and connects to the top with d1 or a3, since a3 forms [[Edge template III2d]].

If Blue plays d2, then Red plays c3. This has a bridge down, and connects to the top for the same reason as without the a5-b4 exchange.

The e2-d4 bridge is [[Bolstered template|bolstered]] at e3 ([[Equivalent patterns#Edge equivalence|by Blue's edge]]), so e3 is no better for Blue than d3.

Thus, the only remaining case is 4. d3 .

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 5:c4 c5 d5 B b5 e1 2:a5 4:d3"
/>

Against that, Red plays 5. c4 .

This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. Thus Blue plays b3 instead. Red responds with d2, connecting down via c4 or e4, and connecting to the top because a1 escapes the ladder.

Therefore 1. b4 does win for Red.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-02-06T06:38:18Z

Demer: /* Puzzle 5 */ put up solution for puzzle 5

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, bridging to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in [[Solutions to puzzles#Puzzle 2_2|puzzle 2]].

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]]:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with b2. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the d1,b2 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red.

Showing that b4 wins for Red:

<hexboard size="5x5"
contents="R e2 a1 1:b4 B b5 e1"
/>

If Blue plays to the upper-left of the [[short diagonal]] ''other than'' in {c2,b3}, then Red responds with c2 and defends the bridge. c2 connects to the top via d1 or c1 or b2, and connects to the bottom with a very short [[switchback]].

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:c2 5:b3 7:d4 9:d2 B b5 e1 2:a3 4:c3 6:a5 8:c4"
/>

If Blue plays to the lower-right of the [[short diagonal]] ''other than'' at d4, then Red plays a5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. Thus Blue plays b3 instead. Red responds with d2, and a1 escapes that ladder.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:a5 B b5 e1 S blue:(d3 e3 e4 e5 d5 c5 c4)"
/>

Blue must continue with c3, and what's left of this case is checking that e2 connects down no matter which of the above shaded cells was Blue's first move. If that move was neither d3 nor e3, then Red bridges to d4, and connects down with c4 or c5 or d5. Otherwise, Red plays whichever of d3,e3 Blue didn't play. Since Red has a5 and b4, b5 is [[Dead cell|dead]], so Red ladders to b4.

If Blue plays d4, then Red plays c4, capturing a5 and c5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. If Blue plays b3, then Red plays d2, bridging from c4 and laddering to a1.

This leaves the following tries for Blue.

<hexboard size="5x5"
contents="R e2 a1 1:b4 B b5 e1 S d2 c2 c3 b3 a5"
/>

c2 loses to a3, using [[Edge template III2d]]

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:a3 5:d4 7:d2 B b5 e1 2:c2 4:a5 6:c4"
/>

and a [[switchback]], as shown above.

If Blue plays b3, then Red plays d2, and a1 escapes that ladder. Towards the bottom, if Blue plays outside of {c4,c3,d3}, then Red can play c4 and defend the bridge from d2, and if Blue plays outside of {d4,d5,c5}, then Red can play d4 and defend the c5+d5 [[Captured cell|capture]].

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d2 B b5 e1 2:b3"
/>

In the former case, Red connects down with a5 or c5 or e4. In the latter case, Red connects down with e3 or d3 or c3, since c3 threatens c4 and a5. The sets {c4,c3,d3} and {d4,d5,c5} are disjoint, so Red gets at least one of those, and thereby connects down for a win.

If Blue plays c3, then Red plays d4, capturing e2 and b4 to the bottom. Blue must respond in {d2,c2,b3}, since otherwise Red plays c2, connecting down with d2 or b3 and connecting to the top with d1 or c1 or b2. If Blue plays d2 or c2, then Red plays a3, winning with [[Edge template III2d]]. If Blue plays b3, then Red plays d2, and wins by laddering to a1.

If Blue plays d2, then Red plays c3, connecting down with a5 or d4. Towards the top, blue c2 loses to a3, forming [[Edge template III2d]], and Blue's other moves all lose to red c2, connecting with d1 or c1 or b2.

Blue's only remaining try is a5. Red responds to that with d4, [[Captured cell|capturing]] c5 and d5.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 c5 d5 B b5 e1 2:a5"
/>

Red's main threat is d4, laddering to a1.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 c5 d5 5:d2 B b5 e1 2:a5 S a2 b1 b2 c1 c2 d1 d2 d3 e3"
/>

If Blue plays in {a2,b1,b2,c1,d1}, then Red plays c2. This connects to the top with d1 or c1 or b2, and connects down with a [[Interior template#The box|box]].

If Blue plays c2, then Red plays d2. This connects down with a [[Interior template#The mouth or trapezoid|table]], and connects to the top with d1 or a3, since a3 forms [[Edge template III2d]].

If Blue plays d2, then Red plays c3. This has a bridge down, and connects to the top for the same reason as without the a5-b4 exchange.

The e2-d4 bridge is [[Bolstered template|bolstered]] at e3 ([[Equivalent patterns#Edge equivalence|by Blue's edge]]), so e3 is no better for Blue than d3.

Thus, the only remaining case is 4. d3 .

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 5:c4 c5 d5 B b5 e1 2:a5 4:d3"
/>

Against that, Red plays 5. c4 .

This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. Thus Blue plays b3 instead. Red responds with d2, connecting down via c4 or e4, and connecting to the top because a1 escapes the ladder.

Therefore 1. b4 does win for Red.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-02-06T06:37:32Z

Demer: Undo revision 8425 by Demer (talk)

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, bridging to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in [[Solutions to puzzles#Puzzle 2_2|puzzle 2]].

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]]:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with b2. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the d1,b2 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red. Indeed b4 is winning. We leave the verification of this to be done later.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-02-06T06:33:41Z

Demer: /* Puzzle 5 */ put up solution for puzzle 4

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, bridging to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in [[Solutions to puzzles#Puzzle 2_2|puzzle 2]].

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]]:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with b2. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the d1,b2 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red.

Showing that b4 wins for Red:

<hexboard size="5x5"
contents="R e2 a1 1:b4 B b5 e1"
/>

If Blue plays to the upper-left of the [[short diagonal]] ''other than'' in {c2,b3}, then Red responds with c2 and defends the bridge. c2 connects to the top via d1 or c1 or b2, and connects to the bottom with a very short [[switchback]].

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:c2 5:b3 7:d4 9:d2 B b5 e1 2:a3 4:c3 6:a5 8:c4"
/>

If Blue plays to the lower-right of the [[short diagonal]] ''other than'' at d4, then Red plays a5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. Thus Blue plays b3 instead. Red responds with d2, and a1 escapes that ladder.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:a5 B b5 e1 S blue:(d3 e3 e4 e5 d5 c5 c4)"
/>

Blue must continue with c3, and what's left of this case is checking that e2 connects down no matter which of the above shaded cells was Blue's first move. If that move was neither d3 nor e3, then Red bridges to d4, and connects down with c4 or c5 or d5. Otherwise, Red plays whichever of d3,e3 Blue didn't play. Since Red has a5 and b4, b5 is [[Dead cell|dead]], so Red ladders to b4.

If Blue plays d4, then Red plays c4, capturing a5 and c5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. If Blue plays b3, then Red plays d2, bridging from c4 and laddering to a1.

This leaves the following tries for Blue.

<hexboard size="5x5"
contents="R e2 a1 1:b4 B b5 e1 S d2 c2 c3 b3 a5"
/>

c2 loses to a3, using [[Edge template III2d]]

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:a3 5:d4 7:d2 B b5 e1 2:c2 4:a5 6:c4"
/>

and a [[switchback]], as shown above.

If Blue plays b3, then Red plays d2, and a1 escapes that ladder. Towards the bottom, if Blue plays outside of {c4,c3,d3}, then Red can play c4 and defend the bridge from d2, and if Blue plays outside of {d4,d5,c5}, then Red can play d4 and defend the c5+d5 [[Captured cell|capture]].

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d2 B b5 e1 2:b3"
/>

In the former case, Red connects down with a5 or c5 or e4. In the latter case, Red connects down with e3 or d3 or c3, since c3 threatens c4 and a5. The sets {c4,c3,d3} and {d4,d5,c5} are disjoint, so Red gets at least one of those, and thereby connects down for a win.

If Blue plays c3, then Red plays d4, capturing e2 and b4 to the bottom. Blue must respond in {d2,c2,b3}, since otherwise Red plays c2, connecting down with d2 or b3 and connecting to the top with d1 or c1 or b2. If Blue plays d2 or c2, then Red plays a3, winning with [[Edge template III2d]]. If Blue plays b3, then Red plays d2, and wins by laddering to a1.

If Blue plays d2, then Red plays c3, connecting down with a5 or d4. Towards the top, blue c2 loses to a3, forming [[Edge template III2d]], and Blue's other moves all lose to red c2, connecting with d1 or c1 or b2.

Blue's only remaining try is a5. Red responds to that with d4, [[Captured cell|capturing]] c5 and d5.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 c5 d5 B b5 e1 2:a5"
/>

Red's main threat is d4, laddering to a1.

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 c5 d5 5:d2 B b5 e1 2:a5 S a2 b1 b2 c1 c2 d1 d2 d3 e3"
/>

If Blue plays in {a2,b1,b2,c1,d1}, then Red plays c2. This connects to the top with d1 or c1 or b2, and connects down with a [[Interior template#The box|box]].

If Blue plays c2, then Red plays d2. This connects down with a [[Interior template#The mouth or trapezoid|table]], and connects to the top with d1 or a3, since a3 forms [[Edge template III2d]].

If Blue plays d2, then Red plays c3. This has a bridge down, and connects to the top for the same reason as without the a5-b4 exchange.

The e2-d4 bridge is [[Bolstered template|bolstered]] at e3 ([[Equivalent patterns#Edge equivalence|by Blue's edge]]), so e3 is no better for Blue than d3.

Thus, the only remaining case is 4. d3 .

<hexboard size="5x5"
contents="R e2 a1 1:b4 3:d4 5:c4 c5 d5 B b5 e1 2:a5 4:d3"
/>

Against that, Red plays 5. c4 .

This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with [[Edge template III2d]]. Thus Blue plays b3 instead. Red responds with d2, connecting down via c4 or e4, and connecting to the top because a1 escapes the ladder.

Therefore 1. b4 does win for Red.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-01-31T22:26:54Z

Demer: /* Puzzle 4 */ fixed typo

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, bridging to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in [[Solutions to puzzles#Puzzle 2_2|puzzle 2]].

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]]:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with b2. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the d1,b2 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red. Indeed b4 is winning. We leave the verification of this to be done later.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-01-22T05:08:12Z

Demer: /* Puzzle 4 */ fixed wrong cell

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, briding to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in [[Solutions to puzzles#Puzzle 2_2|puzzle 2]].

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]]:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with b2. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the d1,b2 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red. Indeed b4 is winning. We leave the verification of this to be done later.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-01-21T15:14:41Z

Demer: /* Puzzle 4 */ removed unnecessary word

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, briding to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in [[Solutions to puzzles#Puzzle 2|puzzle 2]].

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]]:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with d1. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the e2,d4 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red. Indeed b4 is winning. We leave the verification of this to be done later.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-01-21T15:13:38Z

Demer: /* Puzzle 4 */ fixed a diagram error and mentioned similarity to puzzle 2

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, briding to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in [[Solutions to puzzles#Puzzle 2|puzzle 2]].

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]] down:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with d1. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the e2,d4 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red. Indeed b4 is winning. We leave the verification of this to be done later.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-01-21T15:06:21Z

Demer: /* Puzzle 4 */ fixed coordinates in one diagram

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, briding to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

showing that d2 wins for Red:

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]] down:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with d1. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
coords="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the e2,d4 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red. Indeed b4 is winning. We leave the verification of this to be done later.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-01-21T15:00:03Z

Demer: /* Puzzle 4 */ put up solution to puzzle 4

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 b1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B d1 E *:(b1 c1) S red:-(e1 area(d1,b3,a3,a1))"
/>
To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via [[edge template IV2i]]. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a [[crescent]], and if b1 is empty, Red plays at a2 and is connected by a different crescent and [[edge template II]]. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.
<hexboard size="5x5"
edges="all"
coords="top left"
contents="R a4 e3 2:d2 B 1:a5 d1 S red:-(e1 area(d1,b3,a3,a1))"
/>
The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:
<hexboard size="5x5"
edges="all"
contents="R a4 e3 2:d2 4:b2 6:b4 B 1:a5 d1 3:e1 5:c2 S red:-(e1 area(d1,b3,a3,a1))"
/>

Thus 1. a5 loses, so Blue's only remaining try is 1. c4.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

Uniqueness:

Blue's main threat is c2, briding to [[edge template III2a]].

<hexboard size="5x5"
contents="R e5 a1 B e1 c1 2:c2 S b1 a2 b2 a3 b3 a4 c2 d2 d1"
/>

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with [[edge template IV2g]]. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

<hexboard size="5x5"
contents="R e5 a1 3:d1 B e1 c1 2:b2 4:c3 S red:(b3 c2)"
/>

Blue continues with c3. This connects right via [[edge template III2d]], and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with [[edge template III2d]] and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

showing that d2 wins for Red:

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1"
/>

The following region is a [[virtual connection]] down:

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2"
/>

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming [[edge template III2d]], and everything else fails to c4, connecting down via one of b5,c5,d4.

Against ''most'' [[Intrusion|intrusions]] in that [[virtual connection]], Red just responds with c3. Specifically, c3 ''almost'' makes this a [[Second order template|second order]] connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
coords="bottom right"
contents="R e5 d2 c3 E *:b5 *:d4"
/>

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming [[edge template III2d]], so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R e5 a1 1:d2 B e1 c1 S d1 c3 b5 d4"
/>

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:c4 7:d3 9:a4 B e1 c1 2:d1 4:b5 6:c3 8:c2"
/>

Before the d1,b2 exchange, Red responds to c3 with d1. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 [[Captured cell|captured]] to the top edge, so Blue must defend against both e3 - forming [[edge template III2d]] - and the [[ladder escape fork]] at c4.

<hexboard size="5x5"
visible="-(a2 c2 line(a1,e1))"
edges="bottom right"
contents="R e5 a1 1:d2 b2 B c3 S d3 c5"
/>

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the e2,d4 exchange, none of Blue's intrusions into

<hexboard size="5x5"
visible="-(e1 area(d1,a1,a4))"
edges="bottom right"
contents="R e5 d2"
/>

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

<hexboard size="5x5"
contents="R e5 a1 1:d2 3:b2 5:a4 B e1 c1 2:d1 4:c2"
/>

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red. Indeed b4 is winning. We leave the verification of this to be done later.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-01-12T04:58:36Z

Demer: /* Puzzle 3 */ hopefully fixed error in my proof

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Red's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the ''*''s in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five ''*''s in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue had played b1 or c1, then that captures c3+b4+a5, as shown below:

<hexboard size="5x5"
visible="-(e1 area(d1,b3,a3,a1))"
edges="bottom right"
coords="partial bottom right"
contents="R a4 e3 d2"
/>

If Blue plays b4, then Red responds with a5. In that case, d2 still connects down - via [[edge template IV2i]] - and that connection [[Dead cell|kills]] b4 and would kill c3 if c3 is blue. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again. Showing that Red can respond to keep ''both'' d2 and b4 connected down will suffice, since in that case, whichever of a5,c3 are blue will be dead.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Thus, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 3:e1 S blue:(b1 c1)"
/>

if c1 is empty, then playing there wins for Red due to c2 or b2, and if b1 is empty, then a2 wins for Red due to a [[crescent]].

If Blue had played a5, then the main part of the work is showing that from the region

<hexboard size="5x5"
visible="-(e1 area(d1,b3,a3,a1))"
edges="bottom right"
coords="partial bottom right"
contents="R a4 e3 d2 B a5"
/>

, Red can keep d2 connected down while ensuring that red b4 would connect down.

Given that, Red wins as follows,

<hexboard size="4x5"
visible="-(c3 area(c4,e4,e2))"
edges="top left"
contents="R a4 e3 2:d2 4:b2 6:b4 B d1 3:e1 5:c2"
/>

so consider the bottom region.

<hexboard size="5x5"
visible="-(e1 area(d1,b3,a3,a1))"
edges="bottom right"
coords="partial bottom right"
contents="R a4 e3 d2 B a5"
/>

Before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Thus 1. a5 loses, so Blue's only remaining try is 1. c4 .

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red threatens

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:c2 B d1 1:c4 S a5 b5 b4 b3 c3 c2 c1 e1 e2 d3 d5 e5 e4"
/>

and

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 S a5 a3 b3 b2 b1 c1 c2 e1 e2 d3 d4 d5 c5"
/>

, so Blue must play one of the cells shaded below.

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S a5 b3 c2 c1 e1 e2 d3 d5"
/>

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

, and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:d4 6:b2 8:b4 10:c3 B d1 1:c4 3:a5 5:e1 7:c2 9:b5"
/>

=== Puzzle 4 ===

The solution to puzzle 4 from this section has not yet been typed up.

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red. Indeed b4 is winning. We leave the verification of this to be done later.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-01-11T22:02:06Z

Demer: /* Eric Demer */ put up solution for puzzle 3

== Piet Hein ==
See [[Solutions to Piet Hein's puzzles]]
== Claude Berge ==
See [[Solutions to Claude Berge's puzzles]]
== Bert Enderton ==
=== Puzzle 1 ===

The unique winning first move is Red b4.
(Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

<hex>R7 C7 Q1
Sd1
Sb3 Hc3 Sd3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via [[Template IIIa]]. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

<hex>R7 C7 Q1
H2d1
H4b2 V7c2 V5e2
V3b3 Hc3 H6d3 Hf3
V1b4 Vc4 Hd4
Hc5 Ve5
Va6</hex>

=== Puzzle 2 ===

A winning move is Red e3. This connects to the top edge via [[Template IIIa]], and to the bottom edge via [[Edge template J5|Template J5]]. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

<hexboard size="7x7"
coords="show"
contents="B c3 E *:d3 R 1:e3 B f5 R b6"
/>

=== Puzzle 3 ===

The unique winning first move is Red c3!
(e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

<hex>R6 C6 Q1 Vf4 Hc2 V1c3</hex>

The following seem like horizontal's (Blue's) best tries from the above position.

{| cellspacing="0" style="width: 500px"
| 1. d2 || e2 || 2. d5 || c5 || 3. d4 || b3 || 4. c4 || a5
|-
| 1. d3 || b3 || 2. b5 || e3
|-
| 1. e1 || d2 || 2. d1 || d1 || 3. c4 || b3 || 4. e3 || a5 || or || 4. b5 || e3
|-
| ........ || .... || 2. c4 || b3 || 3. c5 || e3 || 4. e4 || f3 || 5. e6 || d5
|-
| 1. d5 || b3 || 2. d2 || b2 || 3. c4 || a5 || 4. a6 || c5 || 5. b5 || e3 || 6. d5 || f5
|-
| ........ || .... || ........ || .... || 3. b5 || c5 || 4. c4 || e3
|-
| ........ || .... || ........ || .... || 3. b6 || a6 || 4. b4 || c5 || 5. c4 || e3 || 6. e6 || d4
|-
| 1. c4 || e3 || 2. e2 || b3 || 3. d3 || a5
|-
| 1. b5 || d4 || 2. d3 || f2 || 3. f1 || d2 || 4. c4 || e2 || 5. e4 || e3 || 6. c5 || e5
|-
| ........ || .... || 2. e1 || d2 || 3. d1 || f1 || 4. e2 || f2...
|-
| 1. b4 || d2 || 2. d5 || c5 || 3. c4 || e3 || 4. d4 || e5
|}

=== Puzzle 4 ===

<hexboard size="7x7"
contents="B f4 e4 d4 R f5 f2 1:c4"
/>

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the [[ladder escape]]s at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or [[climbing|climb]] to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a [[ladder escape fork]] by playing in the [[edge template]] between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

== Eric Demer ==

=== Puzzle 1 ===

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 [[domination|capture-dominates]] a4 and a5. Thus Blue plays b4.

<hexboard size="5x5"
contents="R e4 a2 1:a3 B c3 b2 2:b4"
/>

Red responds with e2. This connects to the bottom edge by [[edge template IV2d]], and connects to the top edge as shown.

<hexboard size="5x5"
contents="R e4 a2 1:a3 3:e2 5:d2 7:c2 9:b3 B c3 b2 2:b4 4:e1 6:d1 8:c1"
/>

Uniqueness:

The shaded area is a [[virtual connection]] from c3, so Red must either play in it or play e2.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2 S a5 b4 a4 a3 b2 b3 c2 c3"
/>

The rest of this paragraph uses [[edge template III2a]] extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e4. Blue responds with c4. After that, consider [[edge template III2a]] to the left edge.

<hexboard size="5x5"
contents="R e4 a2 1:e2 B c3 b2 2:c4 E *:b3 S a3 b3 a4 b4 a5 b5"
/>

If Red [[Intrusion|intrudes]] there ''other than'' at ''*'', then Blue responding at ''*'' [[Captured cell|captures]] those 6 cells. If Red [[Intrusion|intrudes]] at ''*'' ''before'' Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even ''threatening'' to disconnect c3 from the left edge, so one gets

<hexboard size="5x5"
contents="R e4 a2 1:e2 3:(d4 e3 d3) 5:d2 7:c2 B c3 b2 2:c4 4:e1 6:d1 8:c1"
/>

At this point c3+c4 escapes the ladder from b2.

=== Puzzle 2 ===

The unique winning move is blue d2.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2"
/>

This connects to the left edge via [[edge template IV2g]], so Red must play either in there or e2. Against ''most'' [[Intrusion|intrusions]] in that template, Blue just responds with c3. Specifically, c3 ''almost'' makes this a [[second order template]], since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

<hexboard size="5x4"
visible="-(a1 area(d3,b5,d5))"
edges="top left"
contents="B a2 d2 c3 E *:a4 *:b2"
/>

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via [[edge template III2b]], Red must play one of the shaded cells in the following diagram.

<hexboard size="5x5"
contents="R d5 e1 B a2 1:d2 S b2 a4 c3 e2"
/>

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to [[Climbing|climb]], for example like this:

<hexboard size="5x5"
contents="R d5 e1 B a2 B 1:d2 R 2:e2 B 3:d4 R 4:a4 B 5:b3 R 6:c3 B 7:c2 R 8:d3 B 9:b5"
/>

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 [[Captured cell|captured]] to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an [[Ladder escape fork|escape fork]] for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the [[edge template IV2g]] work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

<hexboard size="5x5"
contents="R d5 e1 2:e2 4:d3 B a2 1:d2 3:d4 5:b5"
/>

Uniqueness:

Red's main threat is d3, bridging to [[edge template III2d]]. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming [[edge template IV2g]].

If Blue plays c4, then Red plays b4. b4 will be [[Captured cell|captured]] to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

<hexboard size="5x5"
contents="R d5 e1 2:b4 B a2 1:b5"
/>

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, [[Dead cell|killing]] b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue [[Intrusion|intrudes]] in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via [[edge template IV2a]]. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

=== Puzzle 3 ===

The unique winning move is blue b2.

<hexboard size="5x5"
contents="R a4 e3 B d1 1:b2"
/>

That [[Captured cell|captures]] a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is [[Bolstered template|bolstered]] at c1 ([[Equivalent patterns#Edge equivalence|by Blue's edge]]), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via [[edge template IV2e]].

Thus Red plays e1. Blue responds with e2.

<hexboard size="5x5"
contents="R a4 e3 2:e1 B d1 1:b2 3:e2"
/>

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, [[Dead cell|killing]] e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat ''other than'' at c4, then Red can respond at c4, after which Red can keep ''both'' c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the ''*''s in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

Red responds with b2, [[Captured cell|capturing]] b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying ''both'' c4 and a5.

<hexboard size="5x5"
contents="R a4 e3 2:b2 c1 B d1 E *:c4 *:a5"
/>

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via [[edge template IV2i]].

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

<hexboard size="5x5"
contents="R a4 e3 2:b2 4:b4 6:d2 B d1 1:b3 3:c4 5:a3"
/>

Thus, Blue must have played one of the five ''*''s in the following diagram.

<hexboard size="5x5"
contents="R a4 e3 B d1 E *:(c1 b1 b2 c4 a5)"
/>

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue had played b1 or c1, then that captures c3+b4+a5, as shown below:

<hexboard size="5x5"
visible="-(e1 area(d1,b3,a3,a1))"
edges="bottom right"
coords="partial bottom right"
contents="R a4 e3 d2"
/>

If Blue plays b4, then Red responds with a5. In that case, d2 still connects down - via [[edge template IV2i]] - and that connection [[Dead cell|kills]] b4 and would kill c3 if c3 is blue. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again. Showing that Red can respond to keep ''both'' d2 and b4 connected down will suffice, since in that case, whichever of a5,c3 are blue will be dead.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a [[trapezoid]]. Thus, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The other in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 S blue:(b1 c1)"
/>

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that,

<hexboard size="5x5"
contents="R a4 e3 2:d2 c3 b4 a5 B d1 3:e1 S blue:(b1 c1)"
/>

if c1 is empty, then playing there wins for Red due to c2 or b2, and if b1 is empty, then a2 wins for Red due to a [[crescent]].

If Blue had played a5, then the main part of the work is showing that from the region

<hexboard size="5x5"
visible="-(e1 area(d1,b3,a3,a1))"
edges="bottom right"
coords="partial bottom right"
contents="R a4 e3 d2 B a5"
/>

, Red can keep d2 connected down while ensuring that red b4 would connect down.

Given that, Red wins as follows,

<hexboard size="4x5"
visible="-(c3 area(c4,e4,e2))"
edges="top left"
contents="R a4 e3 2:d2 4:b2 6:b4 B d1 3:e1 5:c2"
/>

so consider the bottom region.

<hexboard size="5x5"
visible="-(e1 area(d1,b3,a3,a1))"
edges="bottom right"
coords="partial bottom right"
contents="R a4 e3 d2 B a5"
/>

Before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in [[Edge template IV2i|the template]]. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is [[Bolstered template|bolstered]] at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat ''other than'' at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Thus 1. a5 loses, so Blue's only remaining try is 1. c4 .

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4"
/>

Red must play the shaded region below,

<hexboard size="5x5"
contents="R a4 e3 2:d2 B d1 1:c4 S b1 b2 c1 c2 d2 e1"
/>

since otherwise Red plays b2, capturing that region, after which Blue can't defend against all three of Red's threats.

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 B d1 1:c4 E x:(b3 a3 a5) y:(b5 b4 c3) z:(e2 d3 d4 d5 c5)"
/>

If Blue plays b1 or b2, then Red plays c2, connecting to the top with c1 or e1 and to the bottom with b4 or d4. If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d2. If Blue plays c2 or e1 then Red plays

<hexboard size="5x5"
contents="R a4 e3 2:d2 4:b2 6:a3 8:b4 10:d4 B d1 1:c4 5:b3 7:a5 9:b5 11:c3 S blue:(c2 e1)"
/>

, and then wins with whichever of c2,e1 Blue did not play.

=== Puzzle 4 ===

The solution to puzzle 4 from this section has not yet been typed up.

=== Puzzle 5 ===

To find Red's winning move, let's use [[Mustplay region|mustplay analysis]] to narrow down the possibilities. Blue has several [[threat]]s, which are shown with their respective [[carrier]]s:

[[Bridge]] to [[ziggurat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c2 S d1,area(d2,a2,a5)"
/>

b4 and [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a4)"
/>

b4 and a different [[double threat]]:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:b4 E *:d4 *:c2 S area(a5,c5,e4,e3,c4,d2,d1,a2,a3,e2,a4)"
/>

Ladder and climbing from b2:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:d4 R 2:a5 B 3:b4 R 4:a4 B 5:b2 S area(e4,c5,a5,a2,c1,d1,a4,e3)"
/>

c4 with [[edge template III2b]] and ladder:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 B 1:c4 R 2:d4 B 3:d3 R 4:e3 B 5:d2 S area(a3,a5,e5,e3,d2)"
/>

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a4 B 2:c3 E *:a5,b2,d4,d2"
/>

Red a5 loses to Blue c2 as follows. Apart from [[intrusion|attacking]] Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:
<hexboard size="5x5"
contents="R e2 a1 B b5 e1 R 1:a5 B 2:c2 R 3:a3 B 4:a4 R 5:b3 B 6:b4 E *:c3,d4"
/>

This leaves b4 as the only possible move for Red. Indeed b4 is winning. We leave the verification of this to be done later.

=== Puzzle 6 ===
The unique winning move is c4. In the follow-up, Black must play d3 to get [[edge template IV2b]]::
<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1 R 1:c4 B 2:b3 R 3:a4 B 4:b5 R 5:d3"
/>

=== Puzzle 7 ===
The unique winning move is c6:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4 B 1:c6"
/>
Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a [[ladder escape fork]] from f4. The uniqueness of the winning move is discussed in more detail in the article on the [[Mustplay_region#Solving_Hex_puzzles|mustplay region]].

== Other authors ==

=== Puzzle 1 ===
There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h3 R 2:j2 B 3:i3 R 4:j3 B 5:i4 R 6:j4 B 7:h8 R 8:i6 B 9:g7 E *:e8"
/>

In the second line, Blue has several winning moves starting move 9, but we show just one example.
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:g8 R 2:h7 B 3:h3 R 4:j2 B 5:i3 R 6:j3 B 7:i4 R 8:j4 B 9:i5 R 10:j5
B 11:i6 R 12:j6 B 13:i8 R 14:i7 B 15:f9
"
/>

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:
<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7
B 1:h8 R 2:i4 B 3:i2 R 4:h3 B 5:h2 R 6:g3 B 7:g2 R 8:f3 B 9:f2 R 10:e3
B 11:e2 R 12:c3 B 13:d3 R 14:c4 B 15:d4 R 16:b6 B 17:c7 R 18:a8 E *:d8
"
/>

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

=== Puzzle 2 ===
Note that Red is connected upwards by [[Edge_template_J5|Template J5]]. The only way to prevent Red from connecting downwards is to play in the cell marked (*).
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 E *:b6"
/>
Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing.
If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:c5 B 3:b4"
/>
If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.
<hexboard size="7x7"
coords="show"
contents="R b2 B e3 R b5 B 1:b6 R 2:g4 B 3:d5"
/>,
Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

=== Puzzle 3 ===

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely [[settled region|settled]]. Moreover, i2 [[captured cell|captures]] j1 and j2 and [[dead cell|kills]] h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is [[equivalent patterns|equivalent]] to the following:
<hexboard size="10x10"
coords="show"
contents="B area(b1,b3,j3,j1) R area(a1,a4) B area(a5,a10,c10,d9,d5) R f6 g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>
Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

<hexboard size="10x10"
coords="show"
contents="S red:area(a1,a2) blue:area(b1,b2,c3,e1),e1--j1--j3--h3
blue:area(a5,a10,d8,d7,c6),b10,c10 red:f6,g5
R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9
R 1:e9 3:f8 5:g7 7:i5 9:j6
B 2:e8 4:f7 6:g6 8:h6"
/>

=== Puzzle 4 ===

To simplify the analysis, it helps to note that Blue's g2 [[group]] is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is [[equivalent patterns|equivalent]] to the following:

<hexboard size="10x10"
coords="show"
contents="R area(c6,a8,a10,j10,j6) B area(h1,f3,e5,j5,j1)
R b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
"
/>

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

<hexboard size="10x10"
coords="show"
contents="
S red:area(c6,a8,a10,j10,j6) blue:area(h1,f3,e5,j5,j1)
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
B 1:c4
"
/>
Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main [[threat]]s are 1. g5, connecting via [[Edge template IV1d]] and by the [[ladder escape fork]] 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5
E *:line(g4,h4,j2,j3)
"
/>

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a [[crescent]] and connecting with the assistance of the second row [[ladder escape]] at i2

2. j2 3. h2 4. i2 ([[Dominated_cell#Capture-domination|capture-dominates]] the alternatives i1 & j1) 5. h5 - a double [[ziggurat]] connection

[Note that although we used [[Edge template IV1d]] from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler [[Edge template IV1a]] from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is [[Dominated_cell#Capture-domination|capture-dominated]] by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

=== Puzzle 5 ===
Red's [[mustplay region]] consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 S red:(c4 c5 d4 e3 e4 f2 f3 f4)"
/>
The unique winning move is Red c4. Red then connects for example like this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:c4 B 2:c5 R 3:e4 B 4:d3 R 5:f2 B 6:f1 R 7:e2 B 8:e1 R 9:d2 B 10:d1
R 11:b2 B 12:c2 R 13:b4"
/>

Another move that looks plausible is Red d4, but it fails to this:
<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5 R 1:d4 B 2:e5 R 3:c6 B 4:d3 R 5:c4 B 6:c3 R 7:b4 B 8:b2"
/>

=== Puzzle 6 ===
The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are [[minimax|minimaxing]] moves, while move 7 is a [[foiling]] move that blocks the ladder escape created from Blue 6.
<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10
R 1:b7 B 2:e7 R 3:d7 B 4:d8 R 5:c9 B 6:c10 R 7:b10"
/>

== See Also ==

[[Puzzles|Back to puzzles page]]

[[category:Puzzle]]

Solutions to puzzles

2024-01-06T14:14:01Z

Demer: /* Eric Demer */ put up solutions for puzzles 1 and 2

Puzzles

2024-01-05T12:23:10Z

Demer: /* Puzzle 5 */ fixed wrong color (spotted by bobson)

Solving puzzles is a very good way of becoming a stronger player. Solve as many as possible! And feel free to post your own puzzles here.

== [[Piet Hein]]'s puzzles ==

See article [[Piet Hein's puzzles]]

== [[Claude Berge]]'s puzzles ==

See article [[Claude Berge's puzzles]]

== Bert Enderton ==

=== Puzzle 1 ===

Red to play and win.

<hex>R7 C7 Q1
Hc3 Hf3
Vc4 Hd4
Hc5 Ve5
Va6</hex>

Try it on [https://hexworld.org/board/#7c1,d4:se5f3c4c5a6c3 HexWorld].

=== Puzzle 2 ===

Red to play and win.

<hex>R7 C7 Q1
Hc3
Hf5
Vb6</hex>

Try it on [https://hexworld.org/board/#7c1,c3:sb6f5 HexWorld].

=== Puzzle 3 ===

Red to play and win.

This is a very difficult puzzle whose complete solution is extremely complex.

<hexboard size="6x6"
contents="B c2 R f4"
/>

Try it on [https://hexworld.org/board/#6c1,f4c2 HexWorld].

=== Puzzle 4 ===

Red to play and win.

<hexboard size="7x7"
xcontents="B b4 c4 d4 R b3 b6"
contents="B f4 e4 d4 R f5 f2"
/>

Try it on [https://hexworld.org/board/#7c1,d4:sf2e4f5f4 HexWorld].

== [[User:Demer|Eric Demer]] ==

Despite their small size, the following five 5x5 puzzles are quite difficult: They come from computer brute-force search of sparse 5x5 positions via a tablebase, specifically aiming for the hardest puzzles from such positions. For each of them, the player whose turn it is has exactly 1 winning move.

=== Puzzle 1 ===

Red to move.

<hexboard size="5x5"
contents="R e4 a2 B c3 b2"
/>

Try it on [https://hexworld.org/board/#5c1,e4c3a2b2 HexWorld].

=== Puzzle 2 ===

Blue to move.

<hexboard size="5x5"
contents="R d5 e1 B a2"
/>

Try it on [https://hexworld.org/board/#5c1,d5a2e1 HexWorld].

=== Puzzle 3 ===

Blue to move.

<hexboard size="5x5"
contents="R a4 e3 B d1"
/>

Try it on [https://hexworld.org/board/#5c1,a4d1e3 HexWorld].

=== Puzzle 4 ===

Red to move.

<hexboard size="5x5"
contents="R e5 a1 B e1 c1"
/>

Try it on [https://hexworld.org/board/#5c1,e5e1a1c1 HexWorld].

=== Puzzle 5 ===

Red to move.

<hexboard size="5x5"
contents="R e2 a1 B b5 e1"
/>

Try it on [https://hexworld.org/board/#5c1,e2b5a1e1 HexWorld].

The next two puzzles are based on positions from games.

=== Puzzle 6 ===
Red to play and win.

<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1"
/>

Try it on [https://hexworld.org/board/#6c1,a6:sa3e1b2d1 HexWorld].

=== Puzzle 7 ===
Blue to play and win.

<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4"
/>

Try it on [https://hexworld.org/board/#7c1,a2:pg2c4b5b4e5d7c5d5e3e4g3 HexWorld].

=== Other ===
Eric Demer also has a [[Worst_move_puzzles#Puzzle_3|worst-move puzzle]].

== Other authors ==

=== Puzzle 1 ===
By [[John Tromp]]. Blue to play and win.

<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7"
/>
Try it on [https://hexworld.org/board/#10c1,i1d7c8e6f6f5g5g4h4:pd9 HexWorld].

=== Puzzle 2 ===
By ''lazyplayer''. Blue to play and win.

<hex> R7 C7 Q1
Vb2
He3
Vb5
</hex>

Try it on [https://hexworld.org/board/#7c1,b2e3b5 HexWorld].

=== Puzzle 3 ===
By [[David J Bush]]. Taken from a game on [[Playsite]] in 2003. Red to play and win.

<hexboard size="10x10"
coords="show"
contents="R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>

Try it on [https://hexworld.org/board/#10c1,a3f4e7d9f5c9g4h2h4i2g3g2f3f2e3e2b4b3a4d3c5b5c4d4d5e5e6d6e4g8 HexWorld].

Source: this [[Little Golem]] forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=14 thread].

=== Puzzle 4 ===
Designed by ''Door1'', helped by ''David J Bush''. Inspired by a game on [[Kurnik]] in May 2005. Blue to play and win.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5"
/>

Try it on [https://hexworld.org/board/#10c1,b4b2d7e5d8g2g3f3e4f4c6 HexWorld].

Source: this [[Little Golem]] forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=212 thread].

=== Puzzle 5 ===
By Ryan B. Hayward. Red to play and win.

<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5"
/>

Try it on [https://hexworld.org/board/#6c1,a4b5a6a5 HexWorld].

Note that this position is [[Equivalent patterns|equivalent]] to the position with the pieces at a5 and a6 removed. This could arise in response to the [[Small_boards#Winner_depending_on_the_first_move|winning opening move a4]].

Source: Ryan B. Hayward, "A puzzling Hex primer" (https://webdocs.cs.ualberta.ca/~hayward/papers/puzzlingHexPrimer.pdf).

=== Puzzle 6 ===
By [[User:Hexanna|Hexanna]]. Red to play and win.

<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10"
/>

Try it on [https://hexworld.org/board/#13c1,a3j4d10d4g6g7h5f7i6i3g4g3h3i2c7h8j7j6i7i9k8j10k11l9i10j9h9i8g8h7k9k10 HexWorld].
== See also ==

* [[Worst move puzzles]]
* [[Solutions to puzzles]]
* [[Ladder puzzles]]
* [[Puzzle server]]

[[Cameron Browne]] offers a lot of original puzzles in his book [[Hex Strategy Making the Right Connections]]

[[Matthew Seymour]] has created a website with 500 interactive Hex puzzles at http://www.mseymour.ca/hex_puzzle/hexpuzzle.html

[[Ryan Hayward]] and [[Bjarne Toft]] include several sets of puzzles in their book [[Hex: The Full Story]], including 49 puzzles originally published in Politiken, 28 unpublished puzzles by Jens Lindhard, 99 puzzles by Henderson, and the 4 puzzles by Bert Enderton.

[[category:Puzzle]]

Solutions to puzzles

2024-01-04T18:40:14Z

Demer: I moved 2 solutions since I moved those 2 puzzles.

Puzzles

2024-01-04T18:36:32Z

Demer: I added five 5x5 puzzles, so I made a section for mine.

Solving puzzles is a very good way of becoming a stronger player. Solve as many as possible! And feel free to post your own puzzles here.

== [[Piet Hein]]'s puzzles ==

See article [[Piet Hein's puzzles]]

== [[Claude Berge]]'s puzzles ==

See article [[Claude Berge's puzzles]]

== Bert Enderton ==

=== Puzzle 1 ===

Red to play and win.

<hex>R7 C7 Q1
Hc3 Hf3
Vc4 Hd4
Hc5 Ve5
Va6</hex>

Try it on [https://hexworld.org/board/#7c1,d4:se5f3c4c5a6c3 HexWorld].

=== Puzzle 2 ===

Red to play and win.

<hex>R7 C7 Q1
Hc3
Hf5
Vb6</hex>

Try it on [https://hexworld.org/board/#7c1,c3:sb6f5 HexWorld].

=== Puzzle 3 ===

Red to play and win.

This is a very difficult puzzle whose complete solution is extremely complex.

<hexboard size="6x6"
contents="B c2 R f4"
/>

Try it on [https://hexworld.org/board/#6c1,f4c2 HexWorld].

=== Puzzle 4 ===

Red to play and win.

<hexboard size="7x7"
xcontents="B b4 c4 d4 R b3 b6"
contents="B f4 e4 d4 R f5 f2"
/>

Try it on [https://hexworld.org/board/#7c1,d4:sf2e4f5f4 HexWorld].

== [[User:Demer|Eric Demer]] ==

Despite their small size, the following five 5x5 puzzles are quite difficult: They come from computer brute-force search of sparse 5x5 positions via a tablebase, specifically aiming for the hardest puzzles from such positions. However, for each of them, the player who's turn it is has exactly 1 winning move.

=== Puzzle 1 ===

Red to move

<hexboard size="5x5"
contents="R e4 a2 B c3 b2"
/>

Try it on [https://hexworld.org/board/#5c1,e4c3a2b2 HexWorld].

=== Puzzle 2 ===

Blue to move

<hexboard size="5x5"
contents="R d5 e1 B a2"
/>

Try it on [https://hexworld.org/board/#5c1,d5a2e1 HexWorld].

=== Puzzle 3 ===

Blue to move

<hexboard size="5x5"
contents="R a4 e3 B d1"
/>

Try it on [https://hexworld.org/board/#5c1,a4d1e3 HexWorld].

=== Puzzle 4 ===

Red to move

<hexboard size="5x5"
contents="R e5 a1 B e1 c1"
/>

Try it on [https://hexworld.org/board/#5c1,e5e1a1c1 HexWorld].

=== Puzzle 5 ===

Blue to move

<hexboard size="5x5"
contents="R e2 a1 B b5 e1"
/>

Try it on [https://hexworld.org/board/#5c1,e2b5a1e1 HexWorld].

-

The next two puzzles were based on positions from games.

=== Puzzle 6 ===
By Eric Demer. Red to play and win.

<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1"
/>

Try it on [https://hexworld.org/board/#6c1,a6:sa3e1b2d1 HexWorld].

=== Puzzle 7 ===
By Eric Demer. Blue to play and win.

<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4"
/>

Try it on [https://hexworld.org/board/#7c1,a2:pg2c4b5b4e5d7c5d5e3e4g3 HexWorld].

-

I also have [[https://www.hexwiki.net/index.php/Worst_move_puzzles#Puzzle_3|a worst-move puzzle]].

== Other authors ==

=== Puzzle 1 ===
By [[John Tromp]]. Blue to play and win.

<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7"
/>
Try it on [https://hexworld.org/board/#10c1,i1d7c8e6f6f5g5g4h4:pd9 HexWorld].

=== Puzzle 2 ===
By ''lazyplayer''. Blue to play and win.

<hex> R7 C7 Q1
Vb2
He3
Vb5
</hex>

Try it on [https://hexworld.org/board/#7c1,b2e3b5 HexWorld].

=== Puzzle 3 ===
By [[David J Bush]]. Taken from a game on [[Playsite]] in 2003. Red to play and win.

<hexboard size="10x10"
coords="show"
contents="R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>

Try it on [https://hexworld.org/board/#10c1,a3f4e7d9f5c9g4h2h4i2g3g2f3f2e3e2b4b3a4d3c5b5c4d4d5e5e6d6e4g8 HexWorld].

Source: this [[Little Golem]] forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=14 thread].

=== Puzzle 4 ===
Designed by ''Door1'', helped by ''David J Bush''. Inspired by a game on [[Kurnik]] in May 2005. Blue to play and win.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5"
/>

Try it on [https://hexworld.org/board/#10c1,b4b2d7e5d8g2g3f3e4f4c6 HexWorld].

Source: this [[Little Golem]] forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=212 thread].

=== Puzzle 5 ===
By Ryan B. Hayward. Red to play and win.

<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5"
/>

Try it on [https://hexworld.org/board/#6c1,a4b5a6a5 HexWorld].

Note that this position is [[Equivalent patterns|equivalent]] to the position with the pieces at a5 and a6 removed. This could arise in response to the [[Small_boards#Winner_depending_on_the_first_move|winning opening move a4]].

Source: Ryan B. Hayward, "A puzzling Hex primer" (https://webdocs.cs.ualberta.ca/~hayward/papers/puzzlingHexPrimer.pdf).

=== Puzzle 6 ===
By [[User:Hexanna|Hexanna]]. Red to play and win.

<hexboard size="13x13"
coords="show"
contents="R a3 h3 g4 h5 g6 i6 c7 i7 j7 g8 k8 h9 k9 d10 i10 k11
B i2 g3 i3 d4 j4 j6 f7 g7 h7 h8 i8 i9 j9 l9 j10 k10"
/>

Try it on [https://hexworld.org/board/#13c1,a3j4d10d4g6g7h5f7i6i3g4g3h3i2c7h8j7j6i7i9k8j10k11l9i10j9h9i8g8h7k9k10 HexWorld].
== See also ==

* [[Worst move puzzles]]
* [[Solutions to puzzles]]
* [[Ladder puzzles]]
* [[Puzzle server]]

[[Cameron Browne]] offers a lot of original puzzles in his book [[Hex Strategy Making the Right Connections]]

[[Matthew Seymour]] has created a website with 500 interactive Hex puzzles at http://www.mseymour.ca/hex_puzzle/hexpuzzle.html

[[Ryan Hayward]] and [[Bjarne Toft]] include several sets of puzzles in their book [[Hex: The Full Story]], including 49 puzzles originally published in Politiken, 28 unpublished puzzles by Jens Lindhard, 99 puzzles by Henderson, and the 4 puzzles by Bert Enderton.

[[category:Puzzle]]

Worst move puzzles

2024-01-04T18:08:51Z

Demer: /* Puzzle 3 */ linked to my user-page

Normally, the goal of a puzzle is to find the best move(s) or a winning move. The puzzles on this page have the opposite goal: to find the only ''losing'' move.

The puzzles are approximately sorted by increasing order of difficulty.

=== Puzzle 1 ===

By [[User:Selinger|Selinger]]. Red to play the unique losing move.

<hexboard size="4x4"
contents="R a1 b1 b3 B a2 a3 a4"
/>

Try it on [https://hexworld.org/board/#4c1,a1a2b1a3b3a4 HexWorld].

(Of the 10 empty cells, 9 are winning moves for Red, and 1 is a losing move for Red.)

=== Puzzle 2 ===

By [[User:Selinger|Selinger]]. Red to play the unique losing move.

<hexboard size="4x4"
contents="R a3 B d2"
/>

Try it on [https://hexworld.org/board/#4c1,a3d2 HexWorld].

=== Puzzle 3 ===

By [[User:Demer|Eric Demer]]. Red to play the unique losing move.

<hexboard size="5x5"
contents="R d2 d1 B e1 b1"
/>

Try it on [https://hexworld.org/board/#5c1,d2b1d1e1 HexWorld].

=== Puzzle 4 ===

By [[User:Hexanna|Hexanna]]. Red to play the unique losing move.

<hexboard size="6x6"
contents="R a5 d3 B d1 f4"
/>

Try it on [https://hexworld.org/board/#6c1,a5f4d3d1 HexWorld].

=== Puzzle 5 ===

By [[User:Hexanna|Hexanna]]. Red to play the unique losing move.

This is a difficult puzzle.

<hexboard size="7x7"
contents="R a6 e3 B b6 g4"
/>

Try it on [https://hexworld.org/board/#7c1,a6b6e3g4 HexWorld].

== See also ==

* [[Solutions to worst move puzzles]]

[[category:Puzzle]]

Talk:Hex Bibliography

2024-01-02T15:22:33Z

Demer: Is Cameron's book outdated now?

There are plenty of articles related to Hex and Computer Hex. Be bold, and fill in the gaps ;) --[[User:Gregorio|Gregorio]] 12:55, 3 March 2009 (CET)

On BGA, tiyusufaly asked me about Cameron Browne's Hex book: tiyusufaly was considering buying it, and was wondering if it has anything that's neither in Matthew Seymour's e-book nor here on hexwiki. [[User:Demer|Demer]] ([[User talk:Demer|talk]]) 15:22, 2 January 2024 (UTC)

Near ladder escape

2023-11-25T10:27:38Z

Demer: /* D5 does not escape a 4th row ladder */ presumably this "Red" should've been "Blue"

There are a number of [[ladder]] situations where a player does not technically have a [[ladder escape]], but in practice often ends up escaping the ladder anyway. This usually happens because the opponent must play extremely precisely in order to prevent the ladder from escaping, and can easily miss the correct move. In such cases, we may speak of a '''near ladder escape'''.

This pages lists some common near ladder escapes, and how to thwart them.

== C4 does not escape a 5th row ladder ==

A single stone at c4 (or the equivalent cell on the opposite side of the board) does not escape a 5th row ladder, even when there is a certain amount space on the 6th row as shown here:
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="area(a6,k6,k1,f1)"
contents="R i3 1:e2 B d3 f1 g1"
/>

However, there is only one way to prevent the ladder from connecting. Blue must play as follows.
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="area(a6,k6,k1,f1)"
contents="R i3 1:e2 B d3 f1 g1 B 2:e3 R 3:f2 B 4:h4"
/>
In this situation, 2 followed by 4 is the only winning sequence for Blue. The best Red can do is the following, which is not sufficient to connect Red's ladder:
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="area(a6,k6,k1,f1)"
contents="R i3 e2 B d3 f1 g1 B e3 R f2 B h4 R 5:h3 B 6:i4 R 7:g4 B 8:g3 R 9:f3 B 10:e5 R 11:j2 B 12:f4 R 13:h1 B 14:g2"
/>
Note that Red gets a 5th-to-3rd row [[foldback]], so if Red escapes a 3rd row ladder moving left, Red connects.

Also note that Red would be able to connect if the stone to the left of 13 were not occupied. Therefore, with slightly more space on the 6th row, a single stone at c4 actually does escape a 5th row ladder:
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="area(a6,k6,k1,f1)"
contents="R i3 1:e2 B d3 f1"
/>
Conversely, if there is less space on the 6th row, Blue has additional ways of blocking the ladder, such as this:
<hexboard size="6x11"
edges="bottom right"
coords="none"
visible="area(a6,k6,k1,f1)"
contents="R i3 1:e2 B d3 f1 g1 h1 B 2:e3 R 3:f2 B 4:f3 R 5:g2 B 6:h3 R 7:g3 B 8:f5 R 9:g4 B 10:g6"
/>

== D5 does not escape a 4th row ladder ==

A single stone at D5 (or the equivalent cell on the opposite side of the board) does not escape a 4th row ladder, even when the 6th row is empty as shown here. However, the situation is still very threatening. Red gets both a [[foldback]] and a [[switchback]].
<hexboard size="6x12"
edges="bottom right"
coords="none"
visible="area(a6,l6,l1,f1)"
contents="R i2 1:d3 B c4"
/>
In the above situation, Blue's only winning move is to [[ladder handling|push]].
<hexboard size="6x12"
edges="bottom right"
coords="none"
visible="area(a6,l6,l1,f1)"
contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 E x:e4 y:f5 z:d6"
/>
For move 4, Blue has three possible choices: x, y, or z. If Blue plays moves 4 and 6 at y and z (in either order), Red gets a foldback and a switchback, but does not connect outright:
<hexboard size="6x12"
edges="bottom right"
coords="none"
visible="area(a6,l6,l1,f1)"
contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:f5 R 5:f4 B 6:d6 R 7:e5 B 8:e6 R 9:d5 B 10:c6 R 11:h4 B 12:g4 R 13:h2 B 14:f3 R 15:g1"
/>
Note that Blue cannot play move 6 on the 2nd row, or else Red gets a forcing move that allows Red to connect outright:
<hexboard size="6x12"
edges="bottom right"
coords="none"
visible="area(a6,l6,l1,f1)"
contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:f5 R 5:f4 B 6:e5 R 7:e4 B 8:d5 R 9:h4 B 10:g4 R 11:h2"
/>
If Blue plays move 4 at x, then on the next move, Blue again has three possiblities:
<hexboard size="6x12"
edges="bottom right"
coords="none"
visible="area(a6,l6,l1,f1)"
contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:e4 R 5:f3 E x:g4 y:g5 z:e6"
/>
If Blue plays moves 6 and 8 at y and z (in either order), Red gets a foldback and a switchback:
<hexboard size="6x12"
edges="bottom right"
coords="none"
visible="area(a6,l6,l1,f1)"
contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:e4 R 5:f3 B 6:g5 R 7:g4 B 8:e6 R 9:f5 B 10:f6 R 11:e5 B 12:d6 R 13:i4 B 14:h4 R 15:j3 B 16:g3 R 17:h1"
/>
If Blue plays move 6 at x, Red also gets a foldback and switchback:
<hexboard size="6x12"
edges="bottom right"
coords="none"
visible="area(a6,l6,l1,f1)"
contents="R i2 1:d3 B c4 B 2:d4 R 3:e3 B 4:e4 R 5:f3 B 6:g4 R 7:f4 B 8:e6 R 9:e5 B 10:d6 R 11:f5 B 12:f6 R 13:h5 B 14:g5 R 15:h3 B 16:g3 R 17:h1"
/>
In all other cases, Red connects outright.

'''Climbing.''' If Red lacks both a switchback threat and a foldback threat, Red's goal may be to deny Blue a ladder escape in the corner, and to [[climbing|climb]] as far as possible. Red can play as follows:
<hexboard size="9x12"
edges="bottom right"
coords="none"
visible="area(a9,l9,l1,i1)"
contents="R i5 1:d6 B c7 B 2:d7 R 3:e6 B 4:e7 R 5:f6 B 6:g8 R 7:g7 B 8:g6 R 9:f7 B 10:e9 R 11:j7 B 12:i7 R 13:j6 B 14:i6 R 15:j5 B 16:h6 R 17:i4 B 18:h4 R 19:j2"
/>
Or if Blue plays a different move 12, Red can even do this:
<hexboard size="9x12"
edges="bottom right"
coords="none"
visible="area(a9,l9,l1,i1)"
contents="R i5 1:d6 B c7 B 2:d7 R 3:e6 B 4:e7 R 5:f6 B 6:g8 R 7:g7 B 8:g6 R 9:f7 B 10:e9 R 11:j7 B 12:h7 R 13:k4 B 14:j6 R 15:i7 B 16:i6 R 17:l5 B 18:h6 R 19:i4 B 20:h4 R 21:k1"
/>

== Joseki "C" does not escape a 4th row ladder ==

It is fairly common to play the [[Joseki#4th_row_josekis|4th row joseki]] "C", which leaves the following position in an acute corner:
<hexboard size="5x11"
edges="bottom right"
coords="none"
visible="area(a5,k5,k1,e1)"
contents="R h3 i2 B i3"
/>
This position obviously escapes 2nd row ladders. It is perhaps less obvious that it also escapes 3rd row ladders approaching from far enough away:
<hexboard size="5x11"
edges="bottom right"
coords="none"
visible="area(a5,k5,k1,e1)"
contents="R h3 i2 B i3 B b4 R 1:c3 B 2:c4 R 3:d3 B 4:d4 R 5:f2 B 6:e3 R 7:e2 B 8:f4 R 9:f3 B 10:e4 R 11:g5 S area(c3,b5,h5,h3,i2,i1,g1)"
/>
Note that Red is connected by a [[Interior_template#The_span|span]], and the connection only requires the shaded area. The "magic" move is 5. If Red just continues to push on the 3rd row, Red does not connect.

Does the above corner position escape a 4th row ladder? If Blue naively keeps pushing the ladder, then Red does indeed connect:
<hexboard size="5x11"
edges="bottom right"
coords="none"
visible="area(a5,k5,k1,e1)"
contents="R h3 i2 B i3 B c3 R 1:d2 B 2:d3 R 3:e2 B 4:e3 R 5:f2 B 6:f3 R 7:h1"
/>
On the other hand, if Blue [[ladder handling|yields]] at any point, Red connects by [[switchback]], for example like this:
<hexboard size="5x11"
edges="bottom right"
coords="none"
visible="area(a5,k5,k1,e1)"
contents="R h3 i2 B i3 B c3 R 1:d2 B 2:d3 R 3:e2 B 4:e4 R 5:e3 B 6:d4 R 7:g3 B 8:f3 R 9:g1"
/>
Indeed, for a 4th row ladder approaching the corner, there is only one possible Blue move that prevents Red from escaping the ladder. This "magic move" is 4 in the following diagram:
<hexboard size="5x11"
edges="bottom right"
coords="none"
visible="area(a5,k5,k1,e1)"
contents="R h3 i2 B i3 B c3 R 1:d2 B 2:d3 R 3:e2 B 4:f4 R 5:f3 B 6:d5 R 7:j1 B 8:g3 R 9:h1"
/>
Red still gets a [[foldback]] and a [[switchback]]. Instead of 7, Red could have played anywhere in the corner, but since 7 [[captured cell|captures]] the entire corner, it is usually the [[optimal play|best move]] in this situation.

== See also ==

* [[Ladder escape]]
* [[Switchback]]
* [[Foldback]]

[[Category:Advanced Strategy]]
[[Category:Ladder]]

Physical Hex sets

2023-10-14T23:27:54Z

Demer: added new entry to current, and moved one entry from current to historical

== Historial sets ==

* Pencil-and-paper Hex pads were published by [[Piet Hein]] in Demark in 1943 under the name "Polygon". Each pad contained 50 sheets.

* A Hex set was marketed [https://boardgamegeek.com/image/865770/hex under the name "Hex"] by Parker Brothers, starting in 1952.

* In 1968, Piet Hein marketed a Hex set under the name [https://boardgamegeek.com/images/boardgameversion/337244/skjode-skjern-danishenglish-edition Con-Tac-Tix]. It was a [https://boardgamegeek.com/image/358786/hex wooden board made from teak], using pegs that fit into holes as the pieces. The board size was 12x12. It was manufactured in Denmark by Skjøde of Skjern on behalf of Parker Brothers. The set came with a [https://www.hasbro.com/common/instruct/Con-Tac-Tix.PDF booklet] of instructions, which was basically a reprint of Martin Gardner's Scientific American column.

* Hand-made Hex sets were at some point available at [https://www.mattesmedjan.se/ Mattesmedjan] in Sweden.

== Current sets ==

* You can also buy your set at [http://hexboard.com/ HexBoard].
* A portable version is available from [http://www.nestorgames.com/ nestorgames].
* There is also a Hex set with French booklet sold by [http://www.cijm.org CIJM].
* A set that might also be suitable for blind players is available at https://luduscience.com/hex.html.
* 11x11/14x14 at [https://www.thegamecrafter.com/games/hex The Game Crafter].
* https://www.etsy.com/shop/atlantichex/?etsrc=sdt has rhombic (so, not wasting a large amount of space) wooden boards. Currently, two are for 11x11, and one has 11x11 on one side and 13x13 on the other side.

== Building your own set ==

Here are some ideas on building a set:
* Greg Conquest made a boarding using a [http://gregconquest.com/hex.html dry-erase board and magnets].
* Łukasz Rygało submitted [http://www.boardgamegeek.com/image/167362 this board] to [http://www.boardgamegeek.com BoardGameGeek].
* In the city of Alicante we have made [http://www.flickr.com/photos/liopic/1688139952 this board] with steel nuts and color-glass balls. We are looking for red and blue glass balls, though.

You can also print out the [[Printable_boards]], in sizes up to 14x14.

[[Category: Resources]]

Mustplay region

2023-10-04T01:16:59Z

Demer: /* In a game */ open-parens was before sentence, so period goes before close-parens

Informally, a ''mustplay region'' for a player is a set of cells in which the player must move to avoid losing immediately. Mustplay analysis is an important tool for analyzing Hex positions, because it can help narrow down the number of possibilities a player must consider.

== Example ==

Consider the following position, with Blue to move:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3"
/>

To determine Blue's mustplay region, Blue should consider the possible ways in which Red could make a connection if it were ''Red's'' turn. These are called Red's ''[[threat]]s''. Red has (at least) the following threats:

* If Red plays at e4, Red is [[virtual connection|connected]] by two templates, namely [[edge template II]] and [[edge template IV2d]]. The [[carrier]] of Red's connection is the set of all cells that are required for the connection, and is highlighted: <hexboard size="7x7"
coords="show"
edges="all"
contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
R *:e4 S red:(e1--d1--d4--f4 area(f4,f7,c7))"
/>

* If Red plays at e5, then Red is connected via two copies of [[edge template II]] and two [[bridge]]s, as shown: <hexboard size="7x7"
coords="show"
edges="all"
contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
R *:e5 S red:(e1--d1--d4 area(d4,e4,f5,f7,e7,d5))"
/>

* Alternatively, if Red plays at e5, Red is also connected via [[edge template II]] and [[edge template III2e]], as shown: <hexboard size="7x7"
coords="show"
edges="all"
contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
R *:e5 S red:(e1--d1--d4 area(d4,e4,f6,f7,c7))"
/> While the last two connections both use a Blue stone at e5, they have different carriers.

* If Red plays at d5, Red is connected via a 3rd row [[ladder]], using f6 as a [[ladder escape]]. In this case, the carrier consists of the path the ladder will take and the space required for the ladder escape: <hexboard size="7x7"
coords="show"
edges="all"
contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
R *:d5 S red:(e1--d1--d5 area(d5,f5,f7,c7))"
/>

Blue's mustplay region consists of those empty cells that are in the carriers of all of Red's known threats. Therefore, Blue's mustplay region consists of the cells d1, e1, e5, e6, e7, and f7.
<hexboard size="7x7"
coords="show"
edges="all"
contents="R c4 d2 d3 d4 f4 f6 B b3 c3 c5 c6 e2 e3
S blue:(d1,e1,e5,e6,e7,f7)"
/>

Note that this does not mean that all of d1, e1, e5, e6, e7, and f7 are winning moves for Blue, or even that Blue has any winning moves at all. Rather, what it means is that all ''other'' moves are losing. In other words, if Blue has any winning moves at all, they must be in Blue's mustplay region. Blue must now consider each of the six moves d1, e1, e5, e6, e7, and f7 and check if any of them are winning, or barring that, which one of them is least likely to be losing.

To help narrow down Blue's choices even further, it helps to consider [[captured cell|captured]] and [[dominated cell|dominated]] cells. In the above example, d1, e1, e7, and f7 are captured by Red, and therefore, Blue should not play there. This leaves Blue with e5 and e6 as the only possible moves to consider. It so happens that e5 is winning and e6 is losing. Therefore, considering the mustplay region has helped Blue identify the only possible winning move. Blue will play e5 and win the game.

== Definition ==

From the point of view of a player, a ''[[threat]]'' is a [[virtual connection]] between the opponent's board edges that the opponent can create in a single move. The ''carrier'' of the threat is the set of cells (empty or not) that are required for the virtual connection to be valid. The player's mustplay region is determined as follows:

* Identify as many threats as possible.

* Determine the intersection of the carriers of all of these threats.

* With respect to the chosen set of threats, the ''mustplay region'' is the set of empty cells in that intersection.

== Properties ==

* All moves outside a player's mustplay region are losing. Moves within the mustplay region may be winning or losing.

* If a player's mustplay region is empty, the player is losing.

* If there are no winning moves in a player's mustplay region, the player is losing.

* The mustplay region is not unique. By considering more opponent threats, a player may arrive at a smaller mustplay region.

== Example: no winning move ==

If there are no winning moves in a player's mustplay region, the player is losing. To illustrate this, consider the following position, with Blue to move.
<hexboard size="5x5"
coords="show"
edges="all"
contents="R b3 c2 d2 e1 B e2 c3 a4 a3"
/>
Red's main threats are:
* d3, connecting via a [[ziggurat]]: <hexboard size="5x5"
coords="show"
edges="all"
contents="R b3 c2 d2 e1 B e2 c3 a4 a3
R *:d3 S red:(area(e5,b5,d3,e3) d2 e1)"
/>
* b4, connecting via [[edge template II]]: <hexboard size="5x5"
coords="show"
edges="all"
contents="R b3 c2 d2 e1 B e2 c3 a4 a3
R *:b4 S red:(a5--b5--b3--c2--d2--e1)"
/>
* c4, connecting via [[edge template II]] and a [[double threat]]: <hexboard size="5x5"
coords="show"
edges="all"
contents="R b3 c2 d2 e1 B e2 c3 a4 a3
R *:c4 S red:(c5,b5,c4,d3,b4,b3,d2,c2,e1)"
/>
The only empty cell in the carrier of all three threats is b5, hence Blue's mustplay region consists of b5. This means that all moves except possibly b5 are losing for Blue.
<hexboard size="5x5"
coords="show"
edges="all"
contents="R b3 c2 d2 e1 B e2 c3 a4 a3
S blue:(b5)"
/>
Unfortunately for Blue, b5 is also losing, because if Blue plays b5, Red can win as follows:
<hexboard size="5x5"
coords="show"
edges="all"
contents="R b3 c2 d2 e1 B e2 c3 a4 a3
B 1:b5 R 2:b4 B 3:a5 R 4:d4"
/>
Therefore Blue has no winning moves at all and is losing the game.

== Applications ==

=== Foiling ===

Consider the following situation, with Blue to move:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R c2 b4 f2 f5 B c4 d4 d5 g3"
/>

Red's main threats are:

* e4, connecting via [[bridge]]s and a [[ziggurat]]: <hexboard size="7x7"
coords="show"
edges="all"
contents="R c2 b4 f2 f5 B c4 d4 d5 g3
R *:e4 S red:area(f1,e3,e6,d7,g7,g5,f5,f3,g1)"
/>
* a6, connecting via a 2nd row [[ladder]] and [[ladder escape]]: <hexboard size="7x7"
coords="show"
edges="all"
contents="R c2 b4 f2 f5 B c4 d4 d5 g3
R *:a6 S red:area(c1,a5,a7,f7,f5,e5,d6,b6,d1)"
/>
* a6, connecting via a 2nd row [[ladder]] and a slightly different [[ladder escape]]: <hexboard size="7x7"
coords="show"
edges="all"
contents="R c2 b4 f2 f5 B c4 d4 d5 g3
R *:a6 S red:area(c1,a5,a7,g7,g5,f5,e6,b6,d1)"
/>

Therefore, Blue's mustplay region consists of the following 5 cells:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R c2 b4 f2 f5 B c4 d4 d5 g3
S blue:area(e6,f6,d7,e7,f7) E x:e6 y:f6 z:d7 u:e7 v:f7"
/>
Of these, y, z, u, and v are losing: if Blue plays there, Red wins by responding at x. Blue's unique winning move is x. This move is also known as a [[foiling]] move, because it takes away Red's template and Red's ladder escape at the same time.

=== Solving Hex puzzles ===

Consider the following puzzle, due to Eric Demer. Blue to move and win.
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4"
/>
At first, the situation seems confusing here. Blue's central stones neither seem to have a convincing connection to the left edge nor to the right one.

Mustplay analysis helps clarify the situation. First, let's note that Red's e3 and g3 are already very strongly connected to the top edge; Blue cannot gain anything by intruding into that connection. (In fact, Red has [[captured cell|captured]] rows 1–3). We therefore concentrate on the bottom part of the board. Within that region, Red's main threats are:

* d4, connecting via [[edge template III2a]]: <hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4
R *:d4 S red:(area(c7,c5,b5,a6,a7),d4,e3)"
/>

* f5, connecting via [[double threat]] of f6 and a 2nd row [[ladder]] at d6, for which b5 and c5 are a [[ladder escape]]: <hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4
R *:f5 S red:area(g3,g4,f7,e7,e6,d6,c7,a7,a6,b5,c5,d6)-e6
E *:d6 *:f6"
/>
We therefore see that Blue's mustplay region consists of the following six cells:
<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4
S blue:(a6 b6 c6 a7 b7 c7) E x:a6 y:b6 z:c6 u:a7 v:b7 w:c7"
/>
Of these, x, y, u, v, and w are losing: if Blue plays there, Red can respond at z, re-establishing both threats.
The unique winning move for Blue is z. In fact, this is basically a [[foiling]] move.

=== In a game ===

Consider the following position from an actual game, with Red to move:
<hexboard size="13x13"
contents="R l1 d10 B e4 R h6 B i10 R h10 B f7 R g8 B h7 R g7 B i4 R j4 B i5 R j5 B j3 R k3 B i6 R j6 B i8 R j8 B i9 R j10 B i11 R k10 B j9 R l8 B h12 R l11 B k2 R l2 B k7 R g5 B h3 R f4 B g2 R f3 B f2 R e3 B e2 R d3 B d2 R b3 B c3 R b4 B c4 R b5 B c5 R b6 B c6 R b7 B c7 R b8 B c8 R b9 B c10 R c9 B d9 R d8 B e9"
/>
One of Blue's threats is e6, with the carrier as shown:
<hexboard size="13x13"
contents="R l1 d10 B e4 R h6 B i10 R h10 B f7 R g8 B h7 R g7 B i4 R j4 B i5 R j5 B j3 R k3 B i6 R j6 B i8 R j8 B i9 R j10 B i11 R k10 B j9 R l8 B h12 R l11 B k2 R l2 B k7 R g5 B h3 R f4 B g2 R f3 B f2 R e3 B e2 R d3 B d2 R b3 B c3 R b4 B c4 R b5 B c5 R b6 B c6 R b7 B c7 R b8 B c8 R b9 B c10 R c9 B d9 R d8 B e9 B *:e6
S blue:area(a10,a13,c11,c10,d9,e9,f8,f6,c6,c3,d2,g3,h4,i4,h7,h8,i9,j9,m7,m4,i7,i3,h2,d2,c3,c7,e8,c10)"
/>
Another Blue threat is f10, with the following carrier. Note that f10 is connected to Blue's i8 group by double threat at g9 and g11, marked with "*".
<hexboard size="13x13"
contents="R l1 d10 B e4 R h6 B i10 R h10 B f7 R g8 B h7 R g7 B i4 R j4 B i5 R j5 B j3 R k3 B i6 R j6 B i8 R j8 B i9 R j10 B i11 R k10 B j9 R l8 B h12 R l11 B k2 R l2 B k7 R g5 B h3 R f4 B g2 R f3 B f2 R e3 B e2 R d3 B d2 R b3 B c3 R b4 B c4 R b5 B c5 R b6 B c6 R b7 B c7 R b8 B c8 R b9 B c10 R c9 B d9 R d8 B e9 B *:f10
S blue:area(a10,a13,c11,c10,d9,e9,e10,g12,h12,i11,j9,m7,m4,h8,i8,e9,d9,c10)
E *:g9,g11"
/>

Apart from the areas near the two blue edges, the only overlap of these two threats is h8, and indeed, h8 is a winning move for Red. (Red could also start by first [[intrusion|intruding]] in the areas near Blue's edges, say at b12, but this is not necessary.)

=== Verification of templates ===

Mustplay analysis is also useful in the verification of templates. In that context, it is sometimes known as ''template reduction''. For example, consider [[edge template V1a]]:
<hexboard size="5x10"
visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
edges="bottom"
coords="none"
contents="R g1"
/>

To show that the template is valid, we must show that Blue has no way of disconnecting the template's red stone from the edge. We use mustplay analysis to reduce the number of possiblities. Red's main threats are:

* Connecting via a [[bridge]] to [[ziggurat]], in two different ways: <hexboard size="5x10"
visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
edges="bottom"
coords="none"
contents="R g1 R *:f3 S red:area(f2,f3,d5,g5,g2)"
/> <hexboard size="5x10"
visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
edges="bottom"
coords="none"
contents="R g1 R *:f3 S red:area(f2,c5,f5,f3,g2)"
/>

* Connecting via [[template IVa]]: <hexboard size="5x10"
visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
edges="bottom"
coords="none"
contents="R g1 R *:f2 S red:area(e2,c3,a5,g5,g3,f2)"
/>

* Connecting via a [[bridge]] and [[template IVa]]: <hexboard size="5x10"
visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
edges="bottom"
coords="none"
contents="R g1 R *:h2 S red:area(h1,d5,j5,j3)"
/>

* Connecting via a [[bridge]] and [[edge template III1b|template III-1-b]]: <hexboard size="5x10"
visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
edges="bottom"
coords="none"
contents="R g1 R *:f3 S red:area(f2,c5,g5,g2)-e5"
/>

* Connecting via [[template IVb]], in two different ways: <hexboard size="5x10"
visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
edges="bottom"
coords="none"
contents="R g1 R *:f2 S red:area(e2,c3,a5,h5,h3,g2)-e4"
/> <hexboard size="5x10"
visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
edges="bottom"
coords="none"
contents="R g1 R *:g2 S red:area(f2,d3,b5,i5,i3,h2)-f4"
/>
Therefore, Blue's mustplay region consists of only three cells:
<hexboard size="5x10"
visible="area(a5,c3,d3,f1,h1,h2,i2,i3,j3,j5)"
edges="bottom"
coords="none"
contents="R g1 S blue:(f3 f5 d5)"
/>
To finish verifying the template, it then remains to show that each of these three moves is losing for Blue. See the article on [[edge template V1a]] for the details.

=== Computer Hex ===

Mustplay analysis is used in computer Hex to reduce the [[branching factor|number of possibilities]] that must be considered for a player's next move. This drastically reduces the size of the search tree.

== See also ==

* [[AND and OR rules]]
* [[Second order template]]s

== References ==

R. Hayward, Y. Björnsson, M. Johanson, M. Kan, N. Po, and J. van Rijswijck: [http://webdocs.cs.ualberta.ca/~hayward/papers/s7x7hex1.pdf "Solving 7x7 Hex with domination, fill-in, and virtual connections"], ''Theoretical Computer Science'' 349;123–139, 2005.

R. Hayward: [http://webdocs.cs.ualberta.ca/~hayward/papers/s7x7hex1.pdf "A puzzling Hex primer"]. In ''Games of No Chance 3'', Cambridge University Press, 56:151–162, 2009.

[[category:Theory]]
[[category:Intermediate Strategy]]
[[category:Computer Hex]]

Losing play

2023-10-01T07:45:34Z

Demer: /* Strategies for losing play */ added section

'''Losing play''' is when a player knows that they are [[optimal play|theoretically losing]], but continues the game in the hopes that the opponent will make a mistake. This typically happens when the winning player is less skilled than the losing player and may not be able to see the winning strategy.

One example of losing play is a [[handicap|handicap game]], where the stronger player is losing at the start of the game, but hopes that the opponent will make enough mistakes to tilt the game in the stronger player's favor.

== Strategies for losing play ==

Losing play feels very different from regular play, because the player already knows that there is no winning move and therefore has no viable options to choose from. The player might be tempted to [[resigning|resign]]. The objective of losing play is to induce the opponent to make a mistake.

=== Giving them enough rope ===

A common strategy for losing play is to make the situation as complicated as possible, so that the opponent will not be able to analyze it in the amount of time they have. This strategy has the gruesome name "giving them enough rope". The name is derived from the proverb "Give someone enough rope, and they will hang themselves". In Hex, this can take several forms:
* ''Do not simplify.'' Don't play any moves that would unnecessarily simplify the board position and clarify the situation for your opponent. For example, if your opponent has a complicated [[virtual connection]] that uses an intricate web of double threats, don't [[intrusion|intrude]] in the double threats, as it would simplify your opponent's connection. Another example is [[Efficiency#Fast_forwarding|fast forwarding]] a ladder, rather than playing it out.
* ''Give your opponent many potential responses.'' If you play a [[forcing move]], your opponent typically only has one possible response, and you are effectively giving them no choice but to play a winning move. Instead, play a move where the opponent must choose from many potential responses. Hopefully they will pick the wrong one.
* ''Play an unexpected move.'' It often helps to play a move that your opponent did not anticipate. The opponent is forced to analyze your move from scratch, and may not have enough time to complete the analysis. Effectively you are giving them a puzzle that they may fail to solve.

=== Playing a fishing move ===

A fishing move is a move that the opponent could [[foiling|foil]], but they may not know how to do so. If the game is still undecided or you are winning, playing a fishing move is a bad idea, because when the opponent foils, it worsens your position. But in losing play you literally have nothing else to lose, so you may as well try a fishing move. If you do so, try to play it as early as possible, hopefully before your opponent can see through what you are doing.

=== Feinting ===

A ''feint'' is a fake threat, designed to get your opponent to overreact. It can be a move that looks like it might threaten your opponent's connection, without actually doing so. The hope is to goad the opponent into defending the connection anyway, [[irrelevant move|wasting]] a move. Feinting is a form of bluffing.

A feint is different from a fishing move, because a fishing move actually does require a response, but the opponent might respond in a way that is disadvantageous for them. On the other hand, a feint is a pure bluff that the opponent could just [[tenuki|ignore]].

=== Aiming for a free move ===

If your opponent is [[intrusion|intruding]] into one of your [[virtual connection]]s, you can defend it in a way that makes it more complicated, in the hope that your opponent ends up playing a move which does not even ''threaten'' to break that connection. Your opponent doing so basically gives you a free move elsewhere.

This is mainly when the opponent does not realize the [[virtual connection]] exists, so the opponent is likely to continue playing nearby, due to (if the opponent is Blue) thinking "stop Red from connecting here" rather than "gain from Red defending Red's connection here".

== Losing play in computer Hex ==

Most [[computer Hex]] algorithms are not good at losing play. Both neural network algorithms and the more traditional alpha-beta-search algorithms are optimized to find the most promising move from a number of possibilities. However, if all available options are clearly losing, these algorithms do not have a notion of which moves are "more" losing than others. When presented with such a position, most algorithms just make random moves.

[[category: Advanced Strategy]]
[[category: Computer Hex]]

Talk:Strategic advice from KataHex

2023-08-06T03:04:58Z

Demer:

Great update! My suggestion would be to avoid recycling terms such as "efficient" and "useless triangle", even if this is how you call these concepts in your mind. It can only lead to confusion down the road if different people adopt the same terms to mean different things. How about "strong" or "influential" or "effective" instead of "efficient"? And I'm not sure where the triangle metaphor comes from; does that concept necessarily need its own name? [[User:Selinger|Selinger]] ([[User talk:Selinger|talk]]) 14:36, 26 July 2023 (UTC)

regarding "useless triangle": That content is already covered by the Dead Cell article. regarding "efficient": Another option is coming up with a different term or phrase for what's currently in the Efficiency article, though I can't yet think of one. [[User:Demer|Demer]] ([[User talk:Demer|talk]]) 04:00, 27 July 2023 (UTC)

I think of pattern B1 in my article as a "useless triangle" because the 3 red stones are really inefficiently placed. Perhaps "wasteful triangle" is more accurate. Regarding "efficient": To be honest, I think it's slightly odd for "efficiency" to refer to the theoretical notion in the [[Efficiency]] article. (For example, it's strange that "efficiency" of a template is proportional to the number of moves, instead of inversely proportional. The more moves something takes, the less efficient it is.) It's hard to think of a word other than "efficient" that exactly captures the notion I'm referring to in my article. Maybe "coherent" or "harmonious"? [[User:Hexanna|Hexanna]] ([[User talk:Hexanna|talk]]) 00:47, 4 August 2023 (UTC)

The "useless triangle" part of my comment was referring to, we could get rid of that article. (presumably by redirecting it to https://www.hexwiki.net/index.php/Dead_cell#Examples) Also, maybe "Speed of wins" could be the new title for the [[Efficiency]] article. This would of course require some changes to that article's phrasing, but I could try putting a proposal on my talk page. [[User:Demer|Demer]] ([[User talk:Demer|talk]]) 12:02, 4 August 2023 (UTC)

@Demer: I don't think it's a good idea to change existing terminology unless there is a really compelling reason. The concept of "useless triangle" did not originate on HexWiki; for example, I am pretty sure it's in Cameron Browne's book. It also appears in historical (now outdated) strategy advice articles, before the whole theory of capture etc was developed. The triangle in B1 and B2 is not in fact useless, and whether it's wasteful depends on how it came about. It's quite common to get such triangles in the endgame simply because the opponent intrudes into various connections that were originally not wasteful but must be defended. But perhaps "wasteful triangle" is a fine name nonetheless.

In the efficiency article, efficiency refers to the idea of doing something with fewer moves. So counting the number of required moves seems to me like a good way to measure it. I agree that this leads to a situation where a larger number of moves means less efficiency. The article actually says "Smaller numbers are more efficient." I don't see any fundamental difficulty there. If the postal service takes 3 days to deliver my mail, it is less efficient than if they take 1 day to do it. One could define the efficiency of a connection to be the negative of the required number of moves, so that the ziggurat would have an efficiency of -3, but I don't think this would be more intuitive. Calling the number of required moves the "inefficiency". It would just lead to more double negations (finding the least inefficient connection, etc), without adding clarity.

"Coherent" or "harmonious" seem reasonable choices; perhaps also something like "balanced"? [[User:Selinger|Selinger]] ([[User talk:Selinger|talk]]) 16:49, 5 August 2023 (UTC)

oh: I had been assuming that term was just created on hexwiki. [[User:Demer|Demer]] ([[User talk:Demer|talk]]) 03:04, 6 August 2023 (UTC)

Talk:Strategic advice from KataHex

2023-08-04T12:02:21Z

Demer: replied again

Talk:Strategic advice from KataHex

2023-07-27T04:00:25Z

Demer: replied to Selinger

Talk:Worst move puzzles

2023-07-08T07:43:28Z

Demer:

Here's another relatively easy one: Red to play the unique losing move.

<hexboard size="4x4"
contents="R a1 a3 B d1"
/>

I like this slightly less than puzzle 3 because there's an imbalance between red and blue stones; on the other hand, it's slightly more elegant than puzzle 3 because it uses fewer stones overall. The answer is basically the same as puzzle 3, so I think one should not have both.
[[User:Selinger|Selinger]] ([[User talk:Selinger|talk]]) 01:49, 2 July 2023 (UTC)

Also, for the record, here is the 7x7 puzzle that Hexanna added and then removed. It is pretty hard, but I still enjoyed thinking about it. I would be in favor of re-adding it to the page, perhaps just indicating that it is a harder puzzle.

<hexboard size="7x7"
contents="R a6 e3 B b6 g4"
/>

[[User:Selinger|Selinger]] ([[User talk:Selinger|talk]]) 01:49, 2 July 2023 (UTC)

Added it back. [[User:Hexanna|Hexanna]] ([[User talk:Hexanna|talk]]) 02:53, 2 July 2023 (UTC)

For some reason, the diagrams appear in black/white after moving the talk page. Writing a test comment to see if it's fixed. [[User:Hexanna|Hexanna]] ([[User talk:Hexanna|talk]]) 04:58, 2 July 2023 (UTC)

On the subject of renaming the page: in general, the Wiki style is to use [https://en.wikipedia.org/wiki/Wikipedia:Naming_conventions_(plurals) singular titles], so we have [[joseki]] and not [[josekis]]. That is presumably why Eric chose the singular. But I agree that [[puzzles]] and [[worst move puzzles]] are an exception, since the page is about a list of puzzles, not about the concept of a puzzle. [[User:Selinger|Selinger]] ([[User talk:Selinger|talk]]) 15:31, 2 July 2023 (UTC)

I chose the singular because I created this with just one puzzle. [[User:Demer|Demer]] ([[User talk:Demer|talk]]) 20:41, 2 July 2023 (UTC)

I figure these puzzles should be sorted by difficulty.
However, I'm not doing that now, because

I have not yet solved #5, and me editing the Solutions page
would probably result in me seeing the solution from there.

and

One would need to decide which of #3,#4 should come before the other of those two.

. [[User:Demer|Demer]] ([[User talk:Demer|talk]]) 20:54, 2 July 2023 (UTC)

I found puzzle 1 quite easy (following the line of reasoning in the solutions) but was actually stumped by puzzle 4 for a while. I would say that for me, 3 < 1 < 4 < 2 < 5 in difficulty. (I created puzzle 5 with KataHex and don't have a good way to motivate the solution yet.) I suspect some others will find puzzle 4 easier than 1, so 3 < 4 < 1 < 2 < 5? [[User:Hexanna|Hexanna]] ([[User talk:Hexanna|talk]]) 21:08, 2 July 2023 (UTC)

Yes; that seems good. [[User:Demer|Demer]] ([[User talk:Demer|talk]]) 21:11, 2 July 2023 (UTC)

(I gave up, and read the solution rather than solving #5.) [[User:Demer|Demer]] ([[User talk:Demer|talk]]) 07:43, 8 July 2023 (UTC)

Talk:Worst move puzzles

2023-07-02T21:11:43Z

Demer: agreed with suggested order

Talk:Worst move puzzles

2023-07-02T20:54:32Z

Demer: mentioned sorting

Talk:Worst move puzzles

2023-07-02T20:41:45Z

Demer: responded

Puzzles

2023-07-01T07:51:56Z

Demer: /* See also */ put in link to article I just created

Solving puzzles is a very good way of becoming a stronger player. Solve as many as possible! And feel free to post your own puzzles here.

== [[Piet Hein]]'s puzzles ==

See article [[Piet Hein's puzzles]]

== [[Claude Berge]]'s puzzles ==

See article [[Claude Berge's puzzles]]

== Bert Enderton ==

=== Puzzle 1 ===

Red to play and win.

<hex>R7 C7 Q1
Hc3 Hf3
Vc4 Hd4
Hc5 Ve5
Va6</hex>

Try it on [https://hexworld.org/board/#7c1,d4:se5f3c4c5a6c3 HexWorld].

=== Puzzle 2 ===

Red to play and win.

<hex>R7 C7 Q1
Hc3
Hf5
Vb6</hex>

Try it on [https://hexworld.org/board/#7c1,c3:sb6f5 HexWorld].

=== Puzzle 3 ===

Red to play and win.

This is a very difficult puzzle whose complete solution is extremely complex.

<hexboard size="6x6"
contents="B c2 R f4"
/>

Try it on [https://hexworld.org/board/#6c1,f4c2 HexWorld].

=== Puzzle 4 ===

Red to play and win.

<hexboard size="7x7"
xcontents="B b4 c4 d4 R b3 b6"
contents="B f4 e4 d4 R f5 f2"
/>

Try it on [https://hexworld.org/board/#7c1,d4:sf2e4f5f4 HexWorld].

== Other authors ==

=== Puzzle 1 ===
By [[John Tromp]]. Blue to play and win.

<hexboard size="10x10"
coords="show"
contents="R i1 h4 g5 f6 d9 c8 B g4 f5 e6 d7"
/>
Try it on [https://hexworld.org/board/#10c1,i1d7c8e6f6f5g5g4h4:pd9 HexWorld].

=== Puzzle 2 ===
By ''lazyplayer''. Blue to play and win.

<hex> R7 C7 Q1
Vb2
He3
Vb5
</hex>

Try it on [https://hexworld.org/board/#7c1,b2e3b5 HexWorld].

=== Puzzle 3 ===
By [[David J Bush]]. Taken from a game on [[Playsite]] in 2003. Red to play and win.

<hexboard size="10x10"
coords="show"
contents="R a3 e3 f3 g3 a4 b4 c4 e4 g4 h4 c5 d5 f5 e6 e7
B e2 f2 g2 h2 i2 b3 d3 d4 f4 b5 e5 d6 g8 c9 d9"
/>

Try it on [https://hexworld.org/board/#10c1,a3f4e7d9f5c9g4h2h4i2g3g2f3f2e3e2b4a4b3d3c5b5c4d4d5e5e6d6e4g8 HexWorld].

Source: this [[Little Golem]] forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=14 thread].

=== Puzzle 4 ===
Designed by ''Door1'', helped by ''David J Bush''. Inspired by a game on [[Kurnik]] in May 2005. Blue to play and win.

<hexboard size="10x10"
coords="show"
contents="
R g3 b4 e4 c6 d7 d8
B b2 g2 f3 f4 e5"
/>

Try it on [https://hexworld.org/board/#10c1,b4b2d7e5d8g2g3f3e4f4c6 HexWorld].

Source: this [[Little Golem]] forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=212 thread].

=== Puzzle 5 ===
By Ryan B. Hayward. Red to play and win.

<hexboard size="6x6"
coords="show"
contents="R a4 a6 B a5 b5"
/>

Try it on [https://hexworld.org/board/#6c1,a4b5a6a5 HexWorld].

Note that this position is [[Equivalent patterns|equivalent]] to the position with the pieces at a5 and a6 removed. This could arise in response to the [[Small_boards#Winner_depending_on_the_first_move|winning opening move a4]].

Source: Ryan B. Hayward, "A puzzling Hex primer" (https://webdocs.cs.ualberta.ca/~hayward/papers/puzzlingHexPrimer.pdf).

=== Puzzle 6 ===
By Eric Demer. Red to play and win. Red has exactly one winning move.

<hexboard size="6x6"
coords="show"
contents="R a3 b2 B d1 e1 f1"
/>

Try it on [https://hexworld.org/board/#6c1,a6:sa3e1b2d1 HexWorld].

=== Puzzle 7 ===
By Eric Demer. Blue to play and win.

<hexboard size="7x7"
coords="show"
edges="all"
contents="R g3 g2 e5 e3 c5 b5 a2 B d7 e4 d5 c4 b4"
/>

Try it on [https://hexworld.org/board/#7c1,a2:pg2c4b5b4e5d7c5d5e3e4g3 HexWorld].

== See also ==

* [[Worst Move Puzzle]]
* [[Solutions to puzzles]]
* [[Ladder puzzles]]
* [[Puzzle server]]

[[Cameron Browne]] offers a lot of original puzzles in his book [[Hex Strategy Making the Right Connections]]

[[Matthew Seymour]] has created a website with 500 interactive Hex puzzles at http://www.mseymour.ca/hex_puzzle/hexpuzzle.html

[[Ryan Hayward]] and [[Bjarne Toft]] include several sets of puzzles in their book [[Hex: The Full Story]], including 49 puzzles originally published in Politiken, 28 unpublished puzzles by Jens Lindhard, 99 puzzles by Henderson, and the 4 puzzles by Bert Enderton.

[[category:Puzzle]]

Worst move puzzles

2023-07-01T07:50:52Z

Demer: created page (I found this puzzle by computer search.)

Normally, the goal of a puzzle is to find a/the best move(s), or a/the winning/moves. However, this puzzle is for the opposite:

<hexboard size="5x5"
contents="R d2 d1 B e1 b1"
/>

Red to play the only _LOSING_ move.

(There are exactly 21 empty cells. 20 of them are winning moves for Red, and 1 of them is a losing move for Red.)

User:Hexanna

2023-05-03T17:26:16Z

Demer: /* Article ideas */ replied to an article question from Talk:Cornering

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

* A couple two-move openings where KataHex's win rate is very close to 50% on 13×13 through 19×19: [https://hexworld.org/board/#13n,c2e9 c2 5-5] and [https://hexworld.org/board/#13n,a3c11 a3 3-3]
* katahex_model_20220618.bin.gz (I'll call this the "strong" net) appears significantly stronger than the "default" net.
** From several self-play games, the strong net appears (very approximately) 300±100 Elo stronger on 15×15 and 500±150 Elo stronger on 19×19 when playing with 400-1000 visits/move.
** From my tests, the strong net beats the default net >50% of the time when playing as Blue against Red b14 (2-2 obtuse corner) opening on 15×15, and >50% of the time when playing as Blue without swap on 19×19.
* The b4 opening appears to be weaker than all 6 of its neighbors. On a large enough board, b4 could be a losing opening, and the swap map could contain a hole:
<hexboard size="5x4"
coords="show"
edges="top left"
contents="S red:all blue:(a1--d1 a2--d2 a3 b4)"
/>
* A 13×13 swap map, with KataHex's self-play Elo estimate of the swap advantage for each opening. Generated using the strong net, with around 30k 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. Hexes without numbers are unfair openings that confer Blue more than a 300 Elo advantage. For example, the fairest opening is g3 (or g11), which KataHex thinks Blue should swap, leaving Blue with a 51.5% win rate, or 10 Elo.
** Key takeaways: The "common" human openings c2, k2, a10, a13 are all reasonably fair. g3 has become more popular recently, for good reason. b4 is rarely played, but it seems fair enough to be suitable for even high-level human play.
<hexboard size="13x13"
coords="show"
contents="S red:all
blue:(a1--l1 a2--k2 a3 a11)
blue:(b13--m13 c12--m12 m11 m3)
E 239:(d3 j11)
187:(e3 i11)
48:(f3 h11)
10:(g3 g11)
68:(h3 f11)
185:(i3 e11)
158:(j3 d11)
107:(a13 m1)
161:(k2 c12)
258:(d2 j12)
110:(c2 k12)
184:(b2 l12)
189:(a2 m12)
207:(a3 m11)
143:(b4 l10)
226:(b11 l3)
247:(a4 m10)
211:(a6 m8)
219:(a7 m7)
197:(a8 m6)
171:(a9 m5)
158:(a10 m4)
131:(a11 m3)"
/>

==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.

==Article ideas==

* '''Motifs''' — very loosely related to joseki; small local patterns that occur in the middle of the board, usually representing optimal play from at least one side but not necessarily both sides
** Motifs have some notion of '''"local efficiency"''' (not to be confused with [[efficiency]]) — some motifs are, on average, good or bad for a particular player. Strong players anecdotally try to play locally efficient moves on large boards where calculating everything is impractical. It would be useful to have some of these rules of thumb written down. Can be thought of as a generalization of dead/captured cells, where LE(dead cell) = 0, and LE(X) ≤ LE(Y) if Y capture-dominates X.
** Here are some examples. In the first motif, Red 1 is often a weak move. Blue's best response is usually at a, or sometimes at b or c as part of a minimaxing play. But d is rarely (possibly never) the best move, because Red can respond with a, and Blue's central stone is now a dead stone. So, for any reasonable working definition of "local efficiency" LE, we have LE(d) < LE(a), and LE(b) = LE(c) due to symmetry. KataHex suggests that LE(b) < LE(a).

<hexboard size="5x5"
coords="none"
edges="none"
contents="R b3 B c3 R 1:d2 E a:c2 b:b4 c:d3 d:c4"
/>

Sometimes, a player will attempt to minimax by placing two stones adjacent to each other, like the unmarked blue stones below. (This is a common human mistake on 19×19; adjacent stones are typically less locally efficient than stones a bridge apart.) Red has several options, such as the adjacent block (*), though a far block is often possible too. It would be enlightening to know, absent other considerations, which block is the most "efficient" for Red, so that on a large board, Red could play this block without thinking too hard. Of course, in general the best move depends on the other stones on the board, and there's no move that strictly dominates another. The best move may even plausibly be to "[[tenuki|play elsewhere]]." Provisionally, KataHex thinks playing at one of A, or the far block at B, is a better first move for Red.

<hexboard size="4x4"
coords="none"
edges="none"
contents="B b1 c1 E *:c2 A:(a1 d1) B:a4"
/>

----

Cataloging the simplest of motifs

A. If Blue plays adjacent to Red in the middle of a large board with no nearby stones, a good response for Red is to play adjacent to both stones at one of (*):

<hexboard size="2x3"
coords="none"
edges="none"
contents="R 1:b1 B 2:b2 E *:(a2 c1)"
/>

B. If Blue plays a bridge move away, either of Red's moves at (*) is good.

<hexboard size="3x4"
coords="none"
edges="none"
contents="R 1:c1 B 2:b3 E *:(a2 d2)"
/>

----

regarding your "I think that's fair, ... find it useful." comment on the Cornering talk page:

I do think a dedicated article for that sort of advice would be good. The title could just be, "Strategic Advice from Katahex", depending on what you were thinking of putting in it.

[[User:Demer|Demer]] ([[User talk:Demer|talk]]) 17:26, 3 May 2023 (UTC)

Cornering

2023-05-03T05:34:55Z

Demer: /* D3 corner move */ showed usually-better move for the defender

In a [[ladder]] situation when no [[ladder escape]] exists, the [[ladder|attacking player]] can [[ladder]] into a corner and create a "quasi-escape piece" at the very last minute. This play is called '''cornering'''.

=== Pivoting ===

The most common form of cornering is ''pivoting'': when the attacker [[ladder handling|breaks the ladder]] by playing one hex ahead. This usually results in a ladder for the opponent. Example:

<hexboard size="5x10"
coords="hide"
edges="bottom right"
contents="R c1 R b3 B b4 R c3 B c4 R d3 B d4 R e3 B e4 R 1:f3 B 2:f4 R 3:g3 B 4:g4 R 5:i3"
/>

This results in a new [[ladder]], but now the attacking player is [[ladder|defending]] instead.

<hexboard size="5x10"
coords="hide"
edges="bottom right"
contents="R c1 R b3 B b4 R c3 B c4 R d3 B d4 R e3 B e4 R f3 B f4 R g3 B g4 R i3
B 6:h3 R 7:i2 B 8:h2 R 9:i1"
/>

Red could have also cornered earlier, resulting, for example, in a 4th row ladder for Blue.

=== C4 corner move ===

Given enough space, the attacker can sometimes get an outcome that is better than merely turning the ladder, though not as good as a [[switchback]]. For example, starting from a 2nd row ladder, the following maneuver lets Red move towards the center, rather than parallel to Blue's edge:

<hexboard size="7x8"
coords="hide"
edges="bottom right"
contents="R a6 B a7 R 1:b6 B 2:b7 R 3:c6 B 4:c7 R 5:f4"
/>

Blue now has several options, but all of them allow Red to connect towards the center of the board:

<hexboard size="7x8"
coords="hide"
edges="bottom right"
contents="R a6 B a7 R b6 B b7 R c6 B c7 R f4
B 6:e5 R 7:f6 B 8:e6 R 9:g3 E *:f5 *:h4 B 10:e4 R 11:f2"
/>
Note how Red's 9 is connected to the edge via a [[double threat]] at the cells marked "*". If Blue instead plays 6 on the second row, the position transposes to one commonly encountered with the D3 corner move (described below).

<hexboard size="7x8"
coords="hide"
edges="bottom right"
contents="R a6 B a7 R b6 B b7 R c6 B c7 R f4
B 6:d6 R 7:e5 B 8:d5 R 9:e3"
/>
Note that Red's 7 is connected to the edge via [[edge template IV2b]].

=== D3 corner move ===

Another useful corner move for second row ladders is this:

<hexboard size="7x10"
coords="hide"
edges="bottom right"
contents="R a6 b6 1:c6 3:d6 5:g5 B a7 b7 2:c7 4:d7"
/>
This allows the attacker to [[climbing|climb]], even without a [[Switchback#Switchback threat|switchback threat]], as follows:
<hexboard size="7x10"
coords="hide"
edges="bottom right"
contents="R a6 b6 c6 d6 g5 7:e6 9:h4 11:g3 13:f2 B a7 b7 c7 d7 6:f6 8:e7 10:f5 12:e4 S area(a1,a5,c5,e3,e1,a1) f1 g1 h1 h2 i1 i2 j1 j2 i3 j3 j4 E *:d3"
/>
This move has some similarities with [[Tom's move]], although it does not require a [[parallel ladder]]. Note that 5 and 9 are connected to the edge by [[edge template IV2b]]. The shaded area is not required for this - i.e., it can be occupied by Blue - although depending on what's to the left, 13 may again be threatening to connect back to the ladder (most likely by playing at "*").

===Strategic considerations===

The above sequences are useful if one has the goal of climbing to a particular hex, or when space is limited. However, it's not uncommon to have an empty corner with lots of space available, especially on larger boards like 19×19. In these cases, the attacker generally shouldn't be fixated on climbing to a particular hex and should aim to "make the most" of her lack of a ladder escape. There are a few ways to do this. KataHex frequently prefers the following Red 7, followed by move 9 at one of (*):
<hexboard size="8x9"
coords="hide"
edges="bottom right"
contents="R a7 B a8 R 1:b7 B 2:b8 R 3:c7 B 4:c8 R 5:f6 B 6:e7 R 7:g4 B 8:e6 E *:(f5 h2)"
/>

If there are no nearby stones, both moves often result in Red climbing towards the center. For example:

<hexboard size="8x9"
coords="hide"
edges="bottom right"
contents="R a7 B a8 R b7 B b8 R c7 B c8 R f6 B e7 R g4 B e6
R 9:h2 B 10:d5 R 11:e3 B 12:c4 R 13:c5 B 14:d4 R 15:d2"
/>

It's unknown whether these sequences represent "optimal" play in some theoretical sense. However, practically speaking, it's nearly impossible to work out the tactics by hand on large boards. These KataHex sequences are by no means the only ways to climb from an empty corner, but they seem pretty consistently solid choices.

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

Talk:Cornering

2023-05-02T05:36:13Z

Demer: took longer look at long D3 line

I'm not sure whether how Katahex plays on an empty board is relevant to this article. The sequences shown in the article (other than the Katahex one) show how Red can definitely play towards the center, regardless of what else is on the board (as long as the amount of space shown in each sequence is available). They are verified by the dfpn-solver. By contrast, the Katahex sequences are just something Katahex seems to prefer in certain situations; they are not verified, nor is the amount of space known. Perhaps these should go into a separate, more speculative subsection? [[User:Selinger|Selinger]] ([[User talk:Selinger|talk]]) 23:14, 1 May 2023 (UTC)

I only included these sequences (and discarded several other candidates) because I saw them in many different scenarios in KataHex self-play, on larger boards like 19×19 where the corner happens to be empty, and not just on an empty board. I have played around with KataHex for 200-300 hours by now (yes, that's hours a day on average), and I thought there were many places where the wiki's advice seemed outdated. There appears to be a big difference in philosophy; here's my side:
* Currently, many articles only contain things that can be verifiably proven in a mathematical sense. That's okay, but I imagine many people come to the wiki only to try to improve their play.
* As a specific example, I think Blue's move 12 in the first "d3 corner move" example is usually weak; in a real game you should probably play a bridge-move away from move 10. The problem is that while the example might be the provably optimal way to climb to "*", the goal of climbing to "*" doesn't seem particularly useful except in the specific situation where "*" is already connected to the top.
* My goals are largely practically minded — I'm trying to contribute insights that I think are useful in real games (particularly larger boards where relevant advice is lacking in this wiki), where you're trying to reason under uncertainty (because you can't possibly calculate everything on a large board), '''so you need imperfect heuristics.''' It's not uncommon to see a totally empty corner in 19×19, where the KataHex sequences are relevant, and it doesn't matter exactly how much space is needed as long as you're fairly sure it's enough. I would argue that an empty corner is far more common than the situations where climbing to a particular hex at "*" is crucial. The KataHex sequences I added are good even if Blue has a ladder escape like in [https://hexworld.org/board/#15nc1,a14n2c4a15b14b15c14c15d14d15e14e15f14f15g14g15h14h15i14i15l13k14m11 this position].
* There are many other examples/articles ([[climbing]]) where the results hold theoretically, which is awesome, but I personally wouldn't recommend a player look at the scenarios if (s)he wants to get stronger, because "climbing to the nth row" is rarely a useful goal in itself unless you consider other factors. There are also many articles which seem highly outdated but might remain for historical reasons ([[V vs. H game 1]]; most strong players now would recognize 2...i1 as a very weak move because of Red j2).
* Which leads to a meta-question: how high of a standard should there be for wiki contributions? I think most of my contributions have high practical value but low theoretical value.
Sorry for the long reply. If you think some of my contributions aren't additive, I won't be offended if you or others let me know.
[[User:Hexanna|Hexanna]] ([[User talk:Hexanna|talk]]) 01:42, 2 May 2023 (UTC)

I agree with most of that! But perhaps we should draw a distinction between specific facts about tactics and general strategic advice. The articles you mention (like the one on climbing) are all about tactics, and I think there is value to knowing specific facts, like "such-and-such is a 4th row ladder escape", or "Red can get such-and-such switchback", or "Red can climb to the 7th row using this much space". To me, this is qualitatively different from general strategic advice like "in some situations, climbing might not be such a good idea" or "Katahex prefers another move to playing the switchback here". I'm not saying that such strategic advice isn't valuable; just that maybe it shouldn't be mixed together with tactics. [[User:Selinger|Selinger]] ([[User talk:Selinger|talk]]) 05:03, 2 May 2023 (UTC)

Hexanna, have you also seen those lines in self-play on 11x11 , 13x13 , 14x14 , where 11x11 uses the c2 opening or the i10 opening or far-closer-to-50% 2-move openings? My understanding is that those sizes are far more popular, and it could be that what you added only shows up significantly closer to 19x19, [[User:Demer|Demer]] ([[User talk:Demer|talk]]) 02:32, 2 May 2023 (UTC)

The vast majority of my self-play experiments have been on size 15 and 19. I haven't seen the KataHex lines on 11×11, unsurprisingly considering the space it requires (and I've not run many games there). It's also rare on 13×13, because the bot consistently likes to take corners early in the game. The closest example I remember is something like [https://hexworld.org/board/#13n,a7d10b8b7b11b12c11e11d11c13d12d13e12e13f12f13g12g13j11i12k9i11j10i10j8h9k6 this]. I agree it shows up much more frequently on size 19, which is a common size on LittleGolem but perhaps less common in "fast" time controls.
[[User:Hexanna|Hexanna]] ([[User talk:Hexanna|talk]]) 02:56, 2 May 2023 (UTC)

I have finally taken more than a passing glance at the long "D3 corner move" line's diagram: Hexanna is of course correct about blue12 there. For that move, anything towards the left other than bridging lets Red get at least switchback. (Even if climbing to "*" was the actual goal, a switchback would suffice for that. However, I think that really, Blue should first play Blue's forcing moves towards the right.) I also agree with the rest of Hexanna's comments about climbing:

I think [[Zipper|zippers]] are different-enough that that section should stay in the Climbing article, although some of the 2nd row ladder and 3rd row ladder Scenarios should probably be moved to a different article. As one data-point for this, I note that I have used Scenario 5 for 2nd row (it was something like [https://hexworld.org/board/#11c1,i5:s:pf8:pg7h5i6j4c11d10d11e10e11g9g10i9h9j7 this]) and Scenario 3 for 3rd row in games, although I don't remember ever using any of the later Scenarios in either of those two sections.

[[User:Demer|Demer]] ([[User talk:Demer|talk]]) 05:36, 2 May 2023 (UTC)

Talk:Cornering

2023-05-02T02:32:13Z

Demer: asked question

User:Wurfmaul

2023-02-21T01:03:37Z

Demer: /* A lemma (as seen on discord!) */ replied

==A lemma (as seen on discord!)==

A: <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(a4, c4, c3, d2, d1)"
contents="R d1 B d2 E x:c2 y:b3 z:c3"
/>
B: <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(a4, c4, c3, d2, d1)"
contents="R d1 c2 B d2 b3"
/>

From blue's perspective, A is at least as good as B.
===Proof===
====When red plays first====
The best red can do in A is to play at x, capturing the two cells below. When red plays at the empty cell in B, it kills the blue cell to the left of it, so the result is equivalent to the result in A.
====When blue plays first====
When blue plays at y in A, the result is at least as good for blue as when blue plays at the empty cell in B. To see this, note that if blue plays at x next, it kills z, and this implies that locally, red's only sensible move is at x, making the position identical to B.

reply by [[User:Demer|Demer]]:

That does not need the top red stone. Also, it still works with a line of 3 red stones instead of the edge. See [[Peep#Automatic_peep|automatic peep]].

User:Wurfmaul

2023-02-21T01:00:29Z

Demer: /* A lemma (as seen on discord!) */

==A lemma (as seen on discord!)==

A: <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(a4, c4, c3, d2, d1)"
contents="R d1 B d2 E x:c2 y:b3 z:c3"
/>
B: <hexboard size="3x4"
float="inline"
edges="bottom"
coords="none"
visible="area(a4, c4, c3, d2, d1)"
contents="R d1 c2 B d2 b3"
/>

From blue's perspective, A is at least as good as B.
===Proof===
====When red plays first====
The best red can do in A is to play at x, capturing the two cells below. When red plays at the empty cell in B, it kills the blue cell to the left of it, so the result is equivalent to the result in A.
====When blue plays first====
When blue plays at y in A, the result is at least as good for blue as when blue plays at the empty cell in B. To see this, note that if blue plays at x next, it kills z, and this implies that locally, red's only sensible move is at x, making the position identical to B.

reply by [[User:Demer|Demer]]:

That does not need the top red stone. (Also, it still works with a line of 3 red stones instead of the edge.)

Current events

2023-01-17T09:47:15Z

Demer: updated with current number of players signed up

A real-time tournament is scheduled to start
at 17:00 UTC, on January 21, on igGameCenter.
The games will be 13x13, and the time control will be

30 minutes, plus 5 seconds per move for each player in each game.

As of 4 days before it starts, there are 8 players signed up for it.

[https://boardgamegeek.com/thread/2995298/hex-monthly-tournament-announcement This link] has further details, including how to register for it.

[[category : hex community]]

Tournaments

2023-01-17T09:46:07Z

Demer: /* igGameCenter tournament starts Jan 21, 2023 */ updated with current number of players signed up

=='''igGameCenter tournament starts Jan 21, 2023'''==
A real-time tournament is scheduled to start
at 17:00 UTC, on January 21, on igGameCenter.
The games will be 13x13, and the time control will be

30 minutes, plus 5 seconds per move for each player in each game.

As of 4 days before it starts, there are 8 players signed up for it.

[https://boardgamegeek.com/thread/2995298/hex-monthly-tournament-announcement This link] has further details, including how to register for it.

==Weekly real time Hex tournament on Ludoteka==
This is a free tournament on an 11x11 grid with a one minute time control.
See [http://www.boardgamegeek.com/thread/680832/weekly-colmena-automated-real-time-hex-tournaments here] for details.

==Online Hex tournament for Human players at Boardspace.net starts Jan 1,2011==
Individual games will be played in real time, but matches will be scheduled by
opponents at an agreed date and time.
More info [http://boardspace.net/cgi-bin/tournament-signup.cgi?tournamentid=23].

==First HEXX6 Tournament in Tiel, The Netherlands at June 6th 2010==

'''Hex, Y, HavannaH, Unlur, Atoll and Pünct'''
More info [http://www.hexboard.com/Tournament.htm here].

==International Tournament 2009 in Granollers (Spain) 2009==

More info [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=353 here].

==Second Spanish Hex 13x13 Online Championship in Little Golem==

The Second Spanish Championship has just started!. More info at [http://spainhex.blogspot.com/ its blog].

== 8th Mind Sports Olympiad in Prague ==

The event will be held from September 27th to October 5th in Prague.
[http://www.deskohrani.cz/cgi/mso/index.pl?telo=propozice.pl&text=uvod.htm&turnaj=oly&hra=hxx&jazyk=en&rok=2008 Hex] might be played depending on the number of participants:

==Spanish Hex 13x13 Online Championship in Little Golem==

The first Spanish Championship started in January 2008, and it was played in [[Little Golem]] in a Round Robin (divided in groups). The winner was [[José María Grau Ribas]], [[user:Gregorio|Gregorio Morales]] finished second while José Ignacio Úbeda ended third. More info at [http://spainhex.blogspot.com/ its blog].

==International Tournament 2006 in Oslo==

Took place on August 11th - 13th 2006. Photos and results can be found on [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=244 littlegolem].

==International Tournament 2005 in Wrocław==

The first international Hex tournament was held in May 2005 in Wrocław, [[Poland]].

Here is some information:
* a [http://masak.org/carl/wroclaw/ blog with results]
* a page with [http://www.photos-wroclaw.prv.pl/ photos from the event]

==Online Team Tournament in 2003==

[[Team Tournament 1]]

==See also==
[[ICGA]]

[[category: hex community]]
[[category: History]]

Tournaments

2022-12-27T05:34:34Z

Demer: added tournament that's not 10 years old :-)

=='''igGameCenter tournament starts Jan 21, 2023'''==
A real-time tournament is scheduled to start
at 17:00 UTC, on January 21, on igGameCenter.
The games will be 13x13, and the time control will be

30 minutes, plus 5 seconds per move for each player in each game.

[https://boardgamegeek.com/thread/2995298/hex-monthly-tournament-announcement This link] has further details, including how to register for it.

==Weekly real time Hex tournament on Ludoteka==
This is a free tournament on an 11x11 grid with a one minute time control.
See [http://www.boardgamegeek.com/thread/680832/weekly-colmena-automated-real-time-hex-tournaments here] for details.

==Online Hex tournament for Human players at Boardspace.net starts Jan 1,2011==
Individual games will be played in real time, but matches will be scheduled by
opponents at an agreed date and time.
More info [http://boardspace.net/cgi-bin/tournament-signup.cgi?tournamentid=23].

==First HEXX6 Tournament in Tiel, The Netherlands at June 6th 2010==

'''Hex, Y, HavannaH, Unlur, Atoll and Pünct'''
More info [http://www.hexboard.com/Tournament.htm here].

==International Tournament 2009 in Granollers (Spain) 2009==

More info [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=353 here].

==Second Spanish Hex 13x13 Online Championship in Little Golem==

The Second Spanish Championship has just started!. More info at [http://spainhex.blogspot.com/ its blog].

== 8th Mind Sports Olympiad in Prague ==

The event will be held from September 27th to October 5th in Prague.
[http://www.deskohrani.cz/cgi/mso/index.pl?telo=propozice.pl&text=uvod.htm&turnaj=oly&hra=hxx&jazyk=en&rok=2008 Hex] might be played depending on the number of participants:

==Spanish Hex 13x13 Online Championship in Little Golem==

The first Spanish Championship started in January 2008, and it was played in [[Little Golem]] in a Round Robin (divided in groups). The winner was [[José María Grau Ribas]], [[user:Gregorio|Gregorio Morales]] finished second while José Ignacio Úbeda ended third. More info at [http://spainhex.blogspot.com/ its blog].

==International Tournament 2006 in Oslo==

Took place on August 11th - 13th 2006. Photos and results can be found on [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=244 littlegolem].

==International Tournament 2005 in Wrocław==

The first international Hex tournament was held in May 2005 in Wrocław, [[Poland]].

Here is some information:
* a [http://masak.org/carl/wroclaw/ blog with results]
* a page with [http://www.photos-wroclaw.prv.pl/ photos from the event]

==Online Team Tournament in 2003==

[[Team Tournament 1]]

==See also==
[[ICGA]]

[[category: hex community]]
[[category: History]]