HexWiki - User contributions [en]

Talk:Wheel

2023-04-18T00:56:44Z

Wurfmaul: 10x10 position

<hex>R5 C5 Vc2 Vd3 Vb4 Hc4</hex>

This pattern is not what I would call a Wheel. It is at most a Broken wheel. I think somebody called it a U-turn. — taral
:Yes, granted. My initial feeling was that it was important to point out that this related position is weak, but maybe it's not important enough to merit mention on this page. We could always create a "U-turn" page, or just refer to the example in the tutorial. [[User:Turing|turing]] 08:18, 10 Feb 2005 (CET)

''Blue should never attempt to intrude by playing at B.'' I believe this is false. Here's a simple example (Blue to move) where Blue's only winning moves are at one of (*). In particular, d2 is winning for Blue, while all of the hexes marked A are losing.

<hexboard size="5x5"
coords="show"
edges="show"
contents="R d1 e2 c3 e5 B c1 b3 b5 E *:(d2 d4) A:(e1 c2 d3)"
/>

Now, this might not be very interesting because playing at d2 just delays the "real" winning d4 move. A more interesting question: is there a position where playing at B is the ''unique'' winning move? This seems trickier than it looks, in many winning positions there's a useless bridge intrusion that also preserves the win for instance, so the move at B isn't unique. Intuitively, I think that in order for the intrusion at B to be better than all intrusions at A, the intrusion at B needs to serve the function of asking a [[question]].

"lazyplayer" from LittleGolem showed me [https://hexworld.org/board/#13n,c2d10j9j4c8g7b10b11l4l3k4k3f8g9i7f11e9d9e7f6i8h7d5e6g8h8c7f7e8e10 a position] loosely based from [https://littlegolem.net/jsp/game/game.jsp?gid=2359962 this game]. I was surprised when I put this in KataHex, because it was the first position I saw where an intrusion at B (specifically, f10) was winning but intrusions at A were losing. But, f10 might not be unique; KataHex thinks c11 or i5 could also win. I played around a bit to see if I could make f10 unique — it was tricky to get KataHex to not think c11 was also winning or close to 50%. Eventually, I found [https://hexworld.org/board/#13n,c2d10j9j4c8g7b10b11l4l3k4k3f8g9i7f11e9d9e7f6i8h7d5e6g8h8c7f7e8e10i5i2i3j2h3d4 a position] where KataHex thinks, after about 50k visits, that f10 is winning (90.3% win rate), and the second best move has only an 8.6% win rate. So it appears that this last position has a unique winning move at B, but I only have KataHex evaluations and not a solid proof. (Note for reproducibility: I used the katahex_model_20220618.bin.gz net with a high value of analysisWideRootNoise, 0.5, to reduce blind spots.)

It would be nice if someone could come up with an example small enough that one can ''prove'' the intrusion at B is the unique winning move. I suspect the smallest example might not even fit on a 5x5 or 6x6 board, since it's a fickle matter getting the condition to hold. [[User:Hexanna|Hexanna]] ([[User talk:Hexanna|talk]]) 01:09, 17 April 2023 (UTC)

[https://hexworld.org/board/#11n,d7b9e7d8e6e8e5f9i6g7i7f6i8g6i5e4g2j2i3j3j5d4j6c4j7b4k5j9k6j10 Here] is a simplification for which the dfpn solver can prove that B is the only winning move. Surely it can be simplified further. --[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 22:07, 17 April 2023 (UTC)
Actually the top-most row and right-most column of this position can simply be discarded to reach a [https://hexworld.org/board/#10n,d6b8e6d7e5e7e4g6i7f8i6j8i5j9i4j2i2j1g1e3j4d3j5c3j6b3d5g5d4f5 10x10 position]. --[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 00:56, 18 April 2023 (UTC)

Talk:Wheel

2023-04-17T22:07:12Z

Wurfmaul: 11x11 position

Little Golem

2023-03-04T10:10:22Z

Wurfmaul: Update, add information on LG tournament types and tournament identifiers.

'''Little Golem''', often abreviated '''LG''', is a [[turn-based]] server maintained by [[Richard Malaschitz]] from [[Wikipedia:Slovakia|Slovakia]]. It offers [[Hex]] on the [[board size|sizes]] 11, 13, 15 and 19, and many of the [[top players]] are playing there.

In fact, the most prestigous title in the Hex world today may very well be the [[Champion]] title for size 13 at Little Golem. As of March 2023, it is currently held by [[lazyplayer]]. The size 19 champion is [[Arek Kulczycki]], and the champion for sizes 11 and 15 is [[Daniel Sepczuk]].

Other games available on Little Golem include [[Go]], [[Chess]], [[Twixt]], [[Havannah]], Reversi, Dvonn, Amazons, Golem's word game, Four in a row, Gomoku, Street Soccer and Dots and Boxes.

== Tournaments ==

=== Tournament types ===
Little Golem offers several types of tournaments.

==== Championship ====
Participants are classified in leagues and play in round-robin tournaments of about nine players each. Depending on one's performance, one can be promoted or demoted for the next edition.

==== Little Golem Cup ====
Previously known as Monthly Cup. The winner of the month is determined in a number of round-robin tournaments. A first round group starts when five players have queued, or when the month ends. Each player can participate in as many first round groups as they like. The winners of the first rounds face each other in the second round. The tournament format allows for more than two rounds for a given month, but as of 2023, in recent years, participation has been too low for that.
Since approximately 2016 (but they have also been retroactively created for previous years), there have been yearly finals for all monthly Cup winners of a given year.

==== INFINITY ====
An eternal ladder which works similar to a Swiss-system tournament.

==== User tournaments ====
Custom user created tournaments. Creating a user tournament requires membership.

=== Deciphering tournament identifiers ===
The identifiers roughly follow the format <board game>.<tournament type>.<game variant>.<tournament edition>.<round>.<group>, but it depends on the tournament type. The tournament types are ch, cv, mc, ld and ut. The game variants for hex are HEX11, DEFAULT (means board size 13), HEX15, HEX19.
The precise formats for hex are
* hex.ch.<tournament edition>.<league>.<group> for size 13 '''ch'''ampionships,
* hex.cv.<game variant>.<tournament edition>.<league>.<group>, where <game variant> is not DEFAULT, for '''v'''ariant '''c'''hampionships (excluding older size 19 ones),
* hex.<game variant>.mc.<year>.<month>.<round>.<group> and
* hex.<game variant>.mc.<year>.final for ('''M'''onthly) '''C'''ups (excluding older size 13 and size 19 ones),
* hex.ld.<game variant> for INFINITY ('''l'''a'''d'''der) tournaments,
* ut.hex.<user tournament id>.<round>.<group> for '''u'''ser '''t'''ournaments
plus the outdated formats
* hex19.ch.<tournament edition>.<league>.<group> for championships for size 19 until the 25th edition, before championships for sizes 11 and 15 were introduced in late 2021, and
* hex19.DEFAULT.mc.<year>.<month>.<round>.<group> for size 19 Monthly Cups from 2016 to 2020,
* hex.mc.<year>.<month>.<round>.<group> and
* hex19.mc.<year>.<month>.<round>.<group> for size 13 respectively size 19 Monthly Cups until 2015.
* hex.in.<game variant>.<round> as an older way to refer to INFINITY tournaments. Old links to infinity tournaments led to very large web pages.

==== Examples ====
* In '''hex.ch.54.3.2''', hex.ch.54 stands for 54th hex championship, 3 stands for third league within that championship, and 2 stands for group 2 within that league. See [https://littlegolem.net/jsp/games/championship.jsp?trnid=hex.ch.54.3.2 Example championship] for an illustration of what leagues and groups are. (Note that, misleadingly, this URL refers to a specific group of the championship, but what is shown is an overview for the whole 54th championship.)
* '''hex.DEFAULT.mc.2022.dec.1.6''' stands for the sixth group of the first round of the December 2022 ('''M'''onthly) '''C'''up. DEFAULT stands for the default board size, which is 13x13. (See [https://littlegolem.net/jsp/games/championship.jsp?trnid=hex.DEFAULT.mc.2022.dec.1.6 Example monthly cup].)

== The Forum ==

One distinctive feature of LG is the forum it provides. Many interesting discussion occur there, and actually [[HexWiki:About|HexWiki]] originates from the forum!

== See also ==

For real-time hex games, [[Kurnik]] is the most popular site.

== External links ==
* The address is [http://www.littlegolem.net littlegolem.net].
*[http://www.edcollins.com/golem/ Ed Collins' great FAQ] on Little Golem

[[category: hex community]]
[[category: other games]]
[[category: online play]]

User:Wurfmaul

2023-02-21T02:31:06Z

Wurfmaul: /* An automatic peep lemma? */ Remove top stone, edit edges to stones, rename

==An automatic peep lemma?==

A: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a4, c4, c3, d2, c2)"
contents="B d2 E x:c2 y:b3 z:c3 R a4--c4"
/>
B: <hexboard size="4x4"
float="inline"
edges="none"
coords="none"
visible="area(a4, c4, c3, d2, c2)"
contents="B d2 b3 R c2 a4--c4"
/>

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]].

Wurfmaul: Thanks for the pointers. I actually thought an edge looks nicer than three stones but it's true stones are more accurate.

User:Wurfmaul

2023-02-20T18:24:09Z

Wurfmaul:

==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.

Talk:Parallelogram boards

2023-01-24T06:54:28Z

Wurfmaul:

== Order of dimensions ==

When I created the original form of this page, I probably decided that ''n'' in ''n×m'' stands for the number of rows based on the convention for matrices. Now I wonder if that was a mistake, since MoHex (and also KataGo) will interpret ''n'' in "boardsize ''n'' ''m''" (which changes the size to ''n×m'' according to [http://webdocs.cs.ualberta.ca/~hayward/hex/WhatYouNeedToKnow.pdf Hex<nowiki>:</nowiki> Passing on the Torch]) as the number of columns. The matrix convention also doesn't match our use of board coordinates: "c" in "c2" determines the column.

So, rather than attempting to fix [[GTP]] by claiming <code>boardsize ''n'' ''m''</code> changes the board size to ''m×n'', I think changing [[Parallelogram boards]] and other pages that followed its(?) convention might be the better solution. --[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 06:48, 24 January 2023 (UTC)

Talk:Parallelogram boards

2023-01-24T06:48:19Z

Wurfmaul: Created page with "== Order of dimensions == When I created this article, I probably decided that ''n'' in ''n×m'' stands for the number of rows based on the convention for matrices. Now I won..."

== Order of dimensions ==

When I created this article, I probably decided that ''n'' in ''n×m'' stands for the number of rows based on the convention for matrices. Now I wonder if that was a mistake, since MoHex (and also KataGo) will interpret ''n'' in "boardsize ''n'' ''m''" (which changes the size to ''n×m'' according to [http://webdocs.cs.ualberta.ca/~hayward/hex/WhatYouNeedToKnow.pdf Hex<nowiki>:</nowiki> Passing on the Torch]) as the number of columns. The matrix convention also doesn't match our use of board coordinates: "c" in "c2" determines the column.

So, rather than attempting to fix [[GTP]] by claiming <code>boardsize ''n'' ''m''</code> changes the board size to ''m×n'', I think changing [[Parallelogram boards]] and other pages that followed its(?) convention might be the better solution. --[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 06:48, 24 January 2023 (UTC)

Small boards

2017-10-05T22:55:04Z

Wurfmaul: updated the preface, removed dead queenbee links

Playing [[Hex]] on [[board]]s of size smaller than 10 × 10 is not very interesting, since many players will be able to play almost perfectly. However it may still be intersting for theoretical studies, and for making [[Puzzles|problems]].

The boards of size up to five can be solved by hand. Hex on 6 × 6 has been solved by [[Queenbee]]. The board sizes 7 to 9 have been solved with computer programs, too.

Here are the winning first moves on the small boards. [[Red (player)|Red]] is vertical and plays first. The [[Hex (board element)|cells]] containing a red [[Piece|stone]] are winning moves for red, while those containing a blue stone are losing.

== Winner depending on the first move ==
The following boards can help you decide where you should [[swap]] when playing on small boards, and it might give you ideas of patterns for bigger boards.
<hex>R2 C2 Q1 Vb1 Va2 Ha1 Hb2</hex>

<hex>R3 C3 Q1 Va2 Va3 Hb1 Vb2 Hb3 Vc1 Vc2 Ha1 Hc3</hex>

<hex>R4 C4 Q1 Va4 Vb3 Vc2 Vd1 Ha1 Ha2 Ha3 Hb1 Hb2 Hb4 Hc1 Hc3 Hc4 Hd2 Hd3 Hd4</hex>

<hex>R5 C5 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Ha3 Hb1 Hc1 Hd1 Hb5 Hc5 Hd5 He5 He4 He3</hex>

<hex>R6 C6 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Va3 Hb1 Hc1 Hd1 He1 Vb5 Vc5 Vd5 Ve5 Ve4 Ve3 Vf1 Vf2 Vf3 Vf4 Hf5 Hf6 He6 Hd6 Hc6 Hb6 Va6</hex>

=== Size 7 ===

Size 7 was first solved by [[Ryan Hayward]] using [[domination]]. The proof tree can be found at http://www.cs.ualberta.ca/~hayward/hex7trees/

<hex>R7 C7 Q1 Ha1 Hb1 Hc1 Hd1 He1 Hf1 Vg1 Ha2 Hb2 Vc2 Hd2 Ve2 Vf2 Vg2 Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Hg3 Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Ha5 Vb5 Vc5 Vd5 Ve5 Vf5 Hg5 Va6 Vb6 Vc6 Hd6 Ve6 Hf6 Hg6 Va7 Hb7 Hc7 Hd7 He7 Hf7 Hg7</hex>

=== Size 8 ===

The outcomes for size 8 were computer generated by [[Javerberg]]. The solution was independently computer generated by Hayward et al. and appeared in [[INJCAI|IJCAI09]].

<hex>
R8 C8 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Vh1
Ha2 Hb2 Hc2 Hd2 He2 Hf2 Vg2 Vh2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Hh3
Ha4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Hh5
Ha6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Hh6
Va7 Vb7 Hc7 Hd7 He7 Hf7 Hg7 Hh7
Va8 Hb8 Hc8 Hd8 He8 Hf8 Hg8 Hh8
</hex>

=== Size 9 ===

The outcomes for size 9 by Jakub Pawlewicz and Ryan Hayward.

<hex>
R9 C9 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Hh1 Vi1
Va2 Vb2 Vc2 Hd2 He2 Hf2 Hg2 Vh2 Vi2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Vh3 Hi3
Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4 Vi4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Vh5 Vi5
Va6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Vh6 Vi6
Ha7 Vb7 Vc7 Vd7 Ve7 Vf7 Vg7 Vh7 Hi7
Va8 Vb8 Hc8 Hd8 He8 Hf8 Vg8 Vh8 Vi8
Va9 Hb9 Hc9 Hd9 He9 Hf9 Hg9 Hh9 Hi9
</hex>

== Reference ==
* This [http://www.ru.is/faculty/yngvi/pdf/HaywardBJKPR05.pdf article] by Ryan Hayward ''et al.'' is a reference for 7x7.
* This [[Little Golem]]'s forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=338 thread] is a reference for size 8x8.
* This [https://webdocs.cs.ualberta.ca/~hayward/papers/pawlhayw.pdf article] by Jakub Pawlewicz and Ryan Hayward is a reference for size 9x9.

== See also ==

* [[Board size]]
* [[Jing Yang]] designed a [[decomposition method]] to find winning strategy in Hex. [http://www.ee.umanitoba.ca/~jingyang/index.html Home Page].
* For corresponding information on the game of Y, please visit [[Where to swap (y)]].

[[Category: Theory]]

Small boards

2017-10-05T22:41:22Z

Wurfmaul: /* Size 7 */ add proof tree link from the top of the page

Playing [[Hex]] on [[board]]s of size smaller than 10 × 10 is not very interesting, since many players will be able to play almost perfectly. However it may still be intersting for theoretical studies, and for making [[Puzzles|problems]].

The boards of size up to five can be solved by hand. Hex on 6 × 6 has been solved by [[Queenbee]].

Here are the winning first moves on the small boards. [[Red (player)|Red]] is vertical and plays first. The [[Hex (board element)|cells]] containing a red [[Piece|stone]] are winning moves for red, while those containing a blue stone are losing. For more details, visit Queenbee's own [http://www.cs.ualberta.ca/~queenbee/openings.html opening page].

''Update:'' The 7 × 7 board has been solved by [[Ryan Hayward|R. Hayward]], et.al. For more details, visit http://www.cs.ualberta.ca/~hayward/hex7trees/

== Winner depending on the first move ==
The following boards can help you decide where you should [[swap]] when playing on small boards, and it might give you ideas of patterns for bigger boards.
<hex>R2 C2 Q1 Vb1 Va2 Ha1 Hb2</hex>

<hex>R3 C3 Q1 Va2 Va3 Hb1 Vb2 Hb3 Vc1 Vc2 Ha1 Hc3</hex>

<hex>R4 C4 Q1 Va4 Vb3 Vc2 Vd1 Ha1 Ha2 Ha3 Hb1 Hb2 Hb4 Hc1 Hc3 Hc4 Hd2 Hd3 Hd4</hex>

<hex>R5 C5 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Ha3 Hb1 Hc1 Hd1 Hb5 Hc5 Hd5 He5 He4 He3</hex>

<hex>R6 C6 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Va3 Hb1 Hc1 Hd1 He1 Vb5 Vc5 Vd5 Ve5 Ve4 Ve3 Vf1 Vf2 Vf3 Vf4 Hf5 Hf6 He6 Hd6 Hc6 Hb6 Va6</hex>

=== Size 7 ===

Size 7 was first solved by [[Ryan Hayward]] using [[domination]]. The proof tree can be found at http://www.cs.ualberta.ca/~hayward/hex7trees/

<hex>R7 C7 Q1 Ha1 Hb1 Hc1 Hd1 He1 Hf1 Vg1 Ha2 Hb2 Vc2 Hd2 Ve2 Vf2 Vg2 Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Hg3 Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Ha5 Vb5 Vc5 Vd5 Ve5 Vf5 Hg5 Va6 Vb6 Vc6 Hd6 Ve6 Hf6 Hg6 Va7 Hb7 Hc7 Hd7 He7 Hf7 Hg7</hex>

=== Size 8 ===

The outcomes for size 8 were computer generated by [[Javerberg]]. The solution was independently computer generated by Hayward et al. and appeared in [[INJCAI|IJCAI09]].

<hex>
R8 C8 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Vh1
Ha2 Hb2 Hc2 Hd2 He2 Hf2 Vg2 Vh2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Hh3
Ha4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Hh5
Ha6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Hh6
Va7 Vb7 Hc7 Hd7 He7 Hf7 Hg7 Hh7
Va8 Hb8 Hc8 Hd8 He8 Hf8 Hg8 Hh8
</hex>

=== Size 9 ===

The outcomes for size 9 by Jakub Pawlewicz and Ryan Hayward.

<hex>
R9 C9 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Hh1 Vi1
Va2 Vb2 Vc2 Hd2 He2 Hf2 Hg2 Vh2 Vi2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Vh3 Hi3
Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4 Vi4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Vh5 Vi5
Va6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Vh6 Vi6
Ha7 Vb7 Vc7 Vd7 Ve7 Vf7 Vg7 Vh7 Hi7
Va8 Vb8 Hc8 Hd8 He8 Hf8 Vg8 Vh8 Vi8
Va9 Hb9 Hc9 Hd9 He9 Hf9 Hg9 Hh9 Hi9
</hex>

== Reference ==
* [[Queenbee]]'s opening [http://www.cs.ualberta.ca/~queenbee/openings.html page] is a reference for sizes under 6x6.
* This [http://www.ru.is/faculty/yngvi/pdf/HaywardBJKPR05.pdf article] by Ryan Hayward ''et al.'' is a reference for 7x7.
* This [[Little Golem]]'s forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=338 thread] is a reference for size 8x8.
* This [https://webdocs.cs.ualberta.ca/~hayward/papers/pawlhayw.pdf article] by Jakub Pawlewicz and Ryan Hayward is a reference for size 9x9.

== See also ==

* [[Board size]]
* [[Jing Yang]] designed a [[decomposition method]] to find winning strategy in Hex. [http://www.ee.umanitoba.ca/~jingyang/index.html Home Page].
* For corresponding information on the game of Y, please visit [[Where to swap (y)]].

[[Category: Theory]]

Small boards

2017-10-05T22:29:32Z

Wurfmaul: /* Size 9 */ changed the attribution

Playing [[Hex]] on [[board]]s of size smaller than 10 × 10 is not very interesting, since many players will be able to play almost perfectly. However it may still be intersting for theoretical studies, and for making [[Puzzles|problems]].

The boards of size up to five can be solved by hand. Hex on 6 × 6 has been solved by [[Queenbee]].

Here are the winning first moves on the small boards. [[Red (player)|Red]] is vertical and plays first. The [[Hex (board element)|cells]] containing a red [[Piece|stone]] are winning moves for red, while those containing a blue stone are losing. For more details, visit Queenbee's own [http://www.cs.ualberta.ca/~queenbee/openings.html opening page].

''Update:'' The 7 × 7 board has been solved by [[Ryan Hayward|R. Hayward]], et.al. For more details, visit http://www.cs.ualberta.ca/~hayward/hex7trees/

== Winner depending on the first move ==
The following boards can help you decide where you should [[swap]] when playing on small boards, and it might give you ideas of patterns for bigger boards.
<hex>R2 C2 Q1 Vb1 Va2 Ha1 Hb2</hex>

<hex>R3 C3 Q1 Va2 Va3 Hb1 Vb2 Hb3 Vc1 Vc2 Ha1 Hc3</hex>

<hex>R4 C4 Q1 Va4 Vb3 Vc2 Vd1 Ha1 Ha2 Ha3 Hb1 Hb2 Hb4 Hc1 Hc3 Hc4 Hd2 Hd3 Hd4</hex>

<hex>R5 C5 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Ha3 Hb1 Hc1 Hd1 Hb5 Hc5 Hd5 He5 He4 He3</hex>

<hex>R6 C6 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Va3 Hb1 Hc1 Hd1 He1 Vb5 Vc5 Vd5 Ve5 Ve4 Ve3 Vf1 Vf2 Vf3 Vf4 Hf5 Hf6 He6 Hd6 Hc6 Hb6 Va6</hex>

=== Size 7 ===

Size 7 was first solved by [[Ryan Hayward]] using [[domination]].

<hex>R7 C7 Q1 Ha1 Hb1 Hc1 Hd1 He1 Hf1 Vg1 Ha2 Hb2 Vc2 Hd2 Ve2 Vf2 Vg2 Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Hg3 Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Ha5 Vb5 Vc5 Vd5 Ve5 Vf5 Hg5 Va6 Vb6 Vc6 Hd6 Ve6 Hf6 Hg6 Va7 Hb7 Hc7 Hd7 He7 Hf7 Hg7</hex>

=== Size 8 ===

The outcomes for size 8 were computer generated by [[Javerberg]]. The solution was independently computer generated by Hayward et al. and appeared in [[INJCAI|IJCAI09]].

<hex>
R8 C8 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Vh1
Ha2 Hb2 Hc2 Hd2 He2 Hf2 Vg2 Vh2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Hh3
Ha4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Hh5
Ha6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Hh6
Va7 Vb7 Hc7 Hd7 He7 Hf7 Hg7 Hh7
Va8 Hb8 Hc8 Hd8 He8 Hf8 Hg8 Hh8
</hex>

=== Size 9 ===

The outcomes for size 9 by Jakub Pawlewicz and Ryan Hayward.

<hex>
R9 C9 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Hh1 Vi1
Va2 Vb2 Vc2 Hd2 He2 Hf2 Hg2 Vh2 Vi2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Vh3 Hi3
Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4 Vi4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Vh5 Vi5
Va6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Vh6 Vi6
Ha7 Vb7 Vc7 Vd7 Ve7 Vf7 Vg7 Vh7 Hi7
Va8 Vb8 Hc8 Hd8 He8 Hf8 Vg8 Vh8 Vi8
Va9 Hb9 Hc9 Hd9 He9 Hf9 Hg9 Hh9 Hi9
</hex>

== Reference ==
* [[Queenbee]]'s opening [http://www.cs.ualberta.ca/~queenbee/openings.html page] is a reference for sizes under 6x6.
* This [http://www.ru.is/faculty/yngvi/pdf/HaywardBJKPR05.pdf article] by Ryan Hayward ''et al.'' is a reference for 7x7.
* This [[Little Golem]]'s forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=338 thread] is a reference for size 8x8.
* This [https://webdocs.cs.ualberta.ca/~hayward/papers/pawlhayw.pdf article] by Jakub Pawlewicz and Ryan Hayward is a reference for size 9x9.

== See also ==

* [[Board size]]
* [[Jing Yang]] designed a [[decomposition method]] to find winning strategy in Hex. [http://www.ee.umanitoba.ca/~jingyang/index.html Home Page].
* For corresponding information on the game of Y, please visit [[Where to swap (y)]].

[[Category: Theory]]

Computer Hex

2017-05-31T20:10:21Z

Wurfmaul: /* Non playing programs */

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

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

== AI techniques used in Hex ==

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

== Programs with AI ==

There are several computer programs which play Hex.

=== Available programs ===

{| class="wikitable"
|-
! Program !! Platforms !! Remarks
|-
| [[MoHex]] || Linux || As of 2010, the strongest available Hex program. It uses the UCT-Monte Carlo approach and is developed at the University of Alberta by Philip Henderson, Broderick Arneson and Ryan Hayward.
|-
| [[Wolve]] || Linux || Gold medallist of 2008 Computer Olympiads.
|-
| [[Six]] || Linux, Unix, Windows || by Gábor Melis.
|-
| [[Hexy]] || Windows || The second strongest program available. It was the first program to use virtual connections and was champion of the 5th Computer Olympiad in 2000.
|-
| [[Hexy (iPhone)|Hexy]] || iPhone || Despite using the same name, this program has no relation to [[Hexy]]. It was released in November 2008, offers an AI opponent; the AI appears to be a custom design and hasn't been rated.
|-
| [https://itunes.apple.com/app/id423845369 Hexatious] || iPad, iPhone || Released in August 2009, appears to offer a stronger AI than the iPhone Hexy app (in particular, Hexatious easily beats the other iPhone app in head-to-head competition).
|-
| [https://itunes.apple.com/app/id397349481 Hex Nash] || iPad, iPhone || Released February 2011, no AI but supports online asynchronous play and local play.
|-
| [http://www.mattesmedjan.se/hexilla/ Hexilla] || Java || By Jonatan Rydh, released in October 2009.
|}

=== Unavailable programs ===

* Mongoose by [[Yngvi Björnsson]], [[Ryan Hayward]], Mike Johanson, Morgan Kan, and Nathan Po.
* Queenbee by [[Jack van Rijswijck]] finished second that year.

== Non playing programs ==

=== Front End ===
* [[HexGui]] is a graphical user interface designed by "ab", mostly used as a front end to play against Six. It is possible however to play against other programs that can communicate via [[GTP]]. It can be downloaded on "ab"'s web [http://mgame99.mg.funpic.de/havannah.php page] (broken link).

=== Reviewing and Editing Programs ===

* [http://canyon23.net/jgame/README_hex.html JHex] by Kevin lets you analyse a game, and databases of games.
* [http://www.drking.plus.com/hexagons/hex/khex.html KHex] by David King is a tool for reviewing games. Very well suited for sharing commented games (it exports games in [[Smart Game Format]]!)

== External link==

=== Articles ===

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

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

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

[[category:Computer Hex]]

Common mistakes

2016-08-06T08:37:57Z

Wurfmaul: /* Wrong variations */ trying to correct my punctuation

== Ladder escaping too early ==

<hex>R6, C10, Q1, Vc6, Ve5, Vg4, Vg3, Vg2, Hd4, Hf3, Hg1, Hh1, Hh3, Hi2, Hb2, Ha6</hex>

Red to move. In this situation Red has a win with perfect play. He only has to find a good [[Ladder escape|ladder escape]] from the [[Ladder|ladder]] starting at h1, g2. c2 is such an escape.

=== Wrong variations ===

However, playing c2 in this stage of the game is a losing move because Blue can play f2 or e2. (Previously this page stated Blue could play f1, too. But Red could answer c3, threatening both b5 and e3.)

<hex>R6 C10 Q1
Hg1 Hh1
Hb2 V1c2 Se2 Sf2 Vg2 Hi2
Hf3 Vg3 Hh3
Hd4 Vg4
Ve5
Vc6 Ha6
</hex>

Another good escape from the [[Second row|second line]] would be c3, but it fails too:

<hex>R6 C10 Q1
Hg1 Hh1
Hb2 V1c3 H2e2 Vg2 Hi2
Hf3 Vg3 Hh3
Hd4 Vg4
Ve5
Vc6 Ha6
</hex>

=== Right variation ===
So what should Red do? The only way to win is to play out the ladder to e2 and ''then'' jump to c2:

<hex>R6 C10 Q1
H4e1 H2f1 Hg1 Hh1
Hb2 V5c2 V3e2 V1f2 Vg2 Hi2
Hf3 Vg3 Hh3
Hd4 Vg4
Ve5
Vc6 Ha6
</hex>

Now Blue cannot block Red's connection. A further development game could be: d2, c3, d3, b5 and Red makes a double [[Bridge|bridge]], connecting c3 and c6.

== Bad [[bridge intrusion]] ==

<hex>R6, C10, Q1, Vc6, Ve5, Vg4, Vg3, Vg2, Hd4, Hf3, Hg1, Hh1, Hh3, Hi2, Hb2, Ha6</hex>

The same situation can be lost if Red intrudes into the d4-f3 bridge at e3. Of course if Blue doesn't see the trap he will fill up the bridge at e4 and then Red's e3 is a ladder escape. But Blue can play f2 and win because the ladder moved from second to the [[Third row|third line]], and Red has no ladder escape from the third line.
''Remember that such a bridge intrusion removes the ladder one line further from the edge, which is often a disadvantage to the attacking player.''

<hex>R6 C10 Q1
Hg1 Hh1
Hb2 H4e2 H2f2 Vg2 Hi2
V1e3 Hf3 Vg3 Hh3
Hd4 V3e4 Vg4
Ve5
Vc6 Ha6
</hex>

[[category:strategy]]

Common mistakes

2016-08-06T08:36:44Z

Wurfmaul: Added a blue stone at a6, removed f1 from the refutations against 1.c2, added a diagram for 1.c3

== Ladder escaping too early ==

<hex>R6, C10, Q1, Vc6, Ve5, Vg4, Vg3, Vg2, Hd4, Hf3, Hg1, Hh1, Hh3, Hi2, Hb2, Ha6</hex>

Red to move. In this situation Red has a win with perfect play. He only has to find a good [[Ladder escape|ladder escape]] from the [[Ladder|ladder]] starting at h1, g2. c2 is such an escape.

=== Wrong variations ===

However, playing c2 in this stage of the game is a losing move because Blue can play f2 or e2 (previously this page stated Blue could play f1, too. But Red could answer c3, threatening both b5 and e3).

<hex>R6 C10 Q1
Hg1 Hh1
Hb2 V1c2 Se2 Sf2 Vg2 Hi2
Hf3 Vg3 Hh3
Hd4 Vg4
Ve5
Vc6 Ha6
</hex>

Another good escape from the [[Second row|second line]] would be c3, but it fails too:

<hex>R6 C10 Q1
Hg1 Hh1
Hb2 V1c3 H2e2 Vg2 Hi2
Hf3 Vg3 Hh3
Hd4 Vg4
Ve5
Vc6 Ha6
</hex>

=== Right variation ===
So what should Red do? The only way to win is to play out the ladder to e2 and ''then'' jump to c2:

<hex>R6 C10 Q1
H4e1 H2f1 Hg1 Hh1
Hb2 V5c2 V3e2 V1f2 Vg2 Hi2
Hf3 Vg3 Hh3
Hd4 Vg4
Ve5
Vc6 Ha6
</hex>

Now Blue cannot block Red's connection. A further development game could be: d2, c3, d3, b5 and Red makes a double [[Bridge|bridge]], connecting c3 and c6.

== Bad [[bridge intrusion]] ==

<hex>R6, C10, Q1, Vc6, Ve5, Vg4, Vg3, Vg2, Hd4, Hf3, Hg1, Hh1, Hh3, Hi2, Hb2, Ha6</hex>

The same situation can be lost if Red intrudes into the d4-f3 bridge at e3. Of course if Blue doesn't see the trap he will fill up the bridge at e4 and then Red's e3 is a ladder escape. But Blue can play f2 and win because the ladder moved from second to the [[Third row|third line]], and Red has no ladder escape from the third line.
''Remember that such a bridge intrusion removes the ladder one line further from the edge, which is often a disadvantage to the attacking player.''

<hex>R6 C10 Q1
Hg1 Hh1
Hb2 H4e2 H2f2 Vg2 Hi2
V1e3 Hf3 Vg3 Hh3
Hd4 V3e4 Vg4
Ve5
Vc6 Ha6
</hex>

[[category:strategy]]

Talk:Theory of ladder escapes

2016-05-29T15:49:19Z

Wurfmaul:

== About third row ladder escapes ==

By definition, a third row ladder escape is a pattern P, for that holds: A+n+P is a template for all n≥0. This equivalent to: A+P and A+1+P are templates.

'''Proof.''' That the condition is necessary is obvious. That it is sufficient follows by induction from the following: For all n≥0, If A+n+P and A+(n+1)+P are templates, then A+(n+2)+P is a template, too. It suffices to consider n=0, because one can afterwards substitute P with (n+P) (and this addition is associative :-)). The proposition is true for n=0, because depending on where blue plays, red can reduce A+2+P to either A+P or A+1+P:
<hexboard size="3x6"
coords="show"
contents="B a1 R b1 E *:e1 E *:f1 B a2 E *:e2 E *:f2 E *:e3 E *:f3"
/>
If blue plays anywhere except b2, a3 or b3, then red plays b2 and connects immediately. If blue plays b2, then red plays c1 and reduces the position to A+1+P. If blue plays a3 or b3, then red first plays at b2, forcing blue to play b3 or a3. Afterwards red plays d1 and reduces the position to A+P. ∎

In practice, in showing that A+1+P is a template, one may assume that blue drops to the first line, because otherwise red can reduce to A+P. This means: 
P is a third row ladder escape if and only if A+P and C+P are templates, where A and C are the patterns
<hexboard size="3x2"
coords="hide"
contents="E *:a1 R b1 E *:a2"
/>
and
<hexboard size="3x2"
coords="hide"
contents="R a1 a2"
/>
. 
Perhaps there is an equivalent proposition which requires even fewer amount of work to decide, I don't know.

For the same reason, i.e. that red may raise the ladder at will, if a pattern P is a third row ladder escape, then D+P is a second row ladder escape, where D is
<hexboard size="3x4"
coords="hide"
contents="E *:a1 +:a2 +:a3 +:d1 +:d2 +:d3"
/>
. 
That means every third row ladder escape is a second row ladder escape, if we move the beginning of the ladder two cells to the left, and assume that these 6 cells in D are empty. The <strike>ladder</strike> latter is natural to assume, since we previously expected a third ladder to come from there.
For example, let P be the following pattern:
<hexboard size="4x4"
coords="hide"
contents="E +:a2 +:a3 +:a4 *:d1 R c1"
/>
With the proposition from above, one can see that this is a third row ladder escape. Therefore, D+P, that is
<hexboard size="4x6"
coords="hide"
contents="E *:a1 *:a2 *:b1 +:a3 +:a4 *:f1 R e1"
/>
is a second row ladder escape. The latter is the same as your last example from the section "2nd row ladder escapes", except with an unnecessary cell on the fourth row.

At the same time,
<hexboard size="4x4"
coords="hide"
contents="E +:a3 +:a4 *:d1 R c1"
/>
and
<hexboard size="4x5"
coords="hide"
contents="E +:a3 +:a4 *:e1 R d1"
/>
are no second row ladder escapes, so in this case it is really necessary to move the start of the ladder by two cells.

--[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 14:25, 29 May 2016 (UTC)

== Two minimaller third row ladder escapes? ==

It appears to me like two of the presented third row ladder escapes are actually not minimal, since the plusses may be moved two cells to the right to obtain
<hexboard size="4x4"
coords="hide"
contents="R b1 E *:d1 E +:a2 E +:a3 E +:a4"
/>
respectively
<hexboard size="6x4"
coords="hide"
contents="E *:a1 E *:b1 R c1 E *:a2 R b2 E +:a4 E +:a5 E +:a6"
/>
. 
--[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 15:13, 29 May 2016 (UTC)

Talk:Theory of ladder escapes

2016-05-29T15:13:25Z

Wurfmaul: /* Two minimaller third row ladder escapes? */

== About third row ladder escapes ==

By definition, a third row ladder escape is a pattern P, for that holds: A+n+P is a template for all n≥0. This equivalent to: A+P and A+1+P are templates.

'''Proof.''' That the condition is necessary is obvious. That it is sufficient follows by induction from the following: For all n≥0, If A+n+P and A+(n+1)+P are templates, then A+(n+2)+P is a template, too. This is true because depending on where blue plays, red can reduce A+(n+2)+P to either A+n+P or A+(n+1)+P. For example, n=0:
<hexboard size="3x6"
coords="show"
contents="B a1 R b1 E *:e1 E *:f1 B a2 E *:e2 E *:f2 E *:e3 E *:f3"
/>
If blue plays anywhere except b2, a3 or b3, then red plays b2 and connects immediately. If blue plays b2, then red plays c1 and reduces the position to A+(n+1)+P. If blue plays a3 or b3, then red first plays at b2, forcing blue to play b3 or a3. Afterwards red plays d1 and reduces the position to A+n+P. ∎

In practice, in showing that A+1+P is a template, one may assume that blue drops to the first line, because otherwise red can reduce to A+P. This means: 
P is a third row ladder escape if and only if A+P and C+P are templates, where A and C are the patterns
<hexboard size="3x2"
coords="hide"
contents="E *:a1 R b1 E *:a2"
/>
and
<hexboard size="3x2"
coords="hide"
contents="R a1 a2"
/>
. 
Perhaps there is an equivalent proposition which requires even fewer amount of work to decide, I don't know.

For the same reason, i.e. that red may raise the ladder at will, if a pattern P is a third row ladder escape, then D+P is a second row ladder escape, where D is
<hexboard size="3x4"
coords="hide"
contents="E *:a1 +:a2 +:a3 +:d1 +:d2 +:d3"
/>
. 
That means every third row ladder escape is a second row ladder escape, if we move the beginning of the ladder two cells to the left, and assume that these 6 cells in D are empty. The <strike>ladder</strike> latter is natural to assume, since we previously expected a third ladder to come from there.
For example, let P be the following pattern:
<hexboard size="4x4"
coords="hide"
contents="E +:a2 +:a3 +:a4 *:d1 R c1"
/>
With the proposition from above, one can see that this is a third row ladder escape. Therefore, D+P, that is
<hexboard size="4x6"
coords="hide"
contents="E *:a1 *:a2 *:b1 +:a3 +:a4 *:f1 R e1"
/>
is a second row ladder escape. The latter is the same as your last example from the section "2nd row ladder escapes", except with an unnecessary cell on the fourth row.

At the same time,
<hexboard size="4x4"
coords="hide"
contents="E +:a3 +:a4 *:d1 R c1"
/>
and
<hexboard size="4x5"
coords="hide"
contents="E +:a3 +:a4 *:e1 R d1"
/>
are no second row ladder escapes, so in this case it is really necessary to move the start of the ladder by two cells.

--[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 14:25, 29 May 2016 (UTC)

== Two minimaller third row ladder escapes? ==

It appears to me like two of the presented third row ladder escapes are actually not minimal, since the plusses may be moved two cells to the right to obtain
<hexboard size="4x4"
coords="hide"
contents="R b1 E *:d1 E +:a2 E +:a3 E +:a4"
/>
respectively
<hexboard size="6x4"
coords="hide"
contents="E *:a1 E *:b1 R c1 E *:a2 R b2 E +:a4 E +:a5 E +:a6"
/>
. 
--[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 15:13, 29 May 2016 (UTC)

Talk:Theory of ladder escapes

2016-05-29T15:12:48Z

Wurfmaul: /* Two minimaller third row ladder escapes? */ new section

== About third row ladder escapes ==

By definition, a third row ladder escape is a pattern P, for that holds: A+n+P is a template for all n≥0. This equivalent to: A+P and A+1+P are templates.

'''Proof.''' That the condition is necessary is obvious. That it is sufficient follows by induction from the following: For all n≥0, If A+n+P and A+(n+1)+P are templates, then A+(n+2)+P is a template, too. This is true because depending on where blue plays, red can reduce A+(n+2)+P to either A+n+P or A+(n+1)+P. For example, n=0:
<hexboard size="3x6"
coords="show"
contents="B a1 R b1 E *:e1 E *:f1 B a2 E *:e2 E *:f2 E *:e3 E *:f3"
/>
If blue plays anywhere except b2, a3 or b3, then red plays b2 and connects immediately. If blue plays b2, then red plays c1 and reduces the position to A+(n+1)+P. If blue plays a3 or b3, then red first plays at b2, forcing blue to play b3 or a3. Afterwards red plays d1 and reduces the position to A+n+P. ∎

In practice, in showing that A+1+P is a template, one may assume that blue drops to the first line, because otherwise red can reduce to A+P. This means: 
P is a third row ladder escape if and only if A+P and C+P are templates, where A and C are the patterns
<hexboard size="3x2"
coords="hide"
contents="E *:a1 R b1 E *:a2"
/>
and
<hexboard size="3x2"
coords="hide"
contents="R a1 a2"
/>
. 
Perhaps there is an equivalent proposition which requires even fewer amount of work to decide, I don't know.

For the same reason, i.e. that red may raise the ladder at will, if a pattern P is a third row ladder escape, then D+P is a second row ladder escape, where D is
<hexboard size="3x4"
coords="hide"
contents="E *:a1 +:a2 +:a3 +:d1 +:d2 +:d3"
/>
. 
That means every third row ladder escape is a second row ladder escape, if we move the beginning of the ladder two cells to the left, and assume that these 6 cells in D are empty. The <strike>ladder</strike> latter is natural to assume, since we previously expected a third ladder to come from there.
For example, let P be the following pattern:
<hexboard size="4x4"
coords="hide"
contents="E +:a2 +:a3 +:a4 *:d1 R c1"
/>
With the proposition from above, one can see that this is a third row ladder escape. Therefore, D+P, that is
<hexboard size="4x6"
coords="hide"
contents="E *:a1 *:a2 *:b1 +:a3 +:a4 *:f1 R e1"
/>
is a second row ladder escape. The latter is the same as your last example from the section "2nd row ladder escapes", except with an unnecessary cell on the fourth row.

At the same time,
<hexboard size="4x4"
coords="hide"
contents="E +:a3 +:a4 *:d1 R c1"
/>
and
<hexboard size="4x5"
coords="hide"
contents="E +:a3 +:a4 *:e1 R d1"
/>
are no second row ladder escapes, so in this case it is really necessary to move the start of the ladder by two cells.

--[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 14:25, 29 May 2016 (UTC)

== Two minimaller third row ladder escapes? ==

It appears to me like two of the presented third row ladder escapes are actually not minimal, since the plusses may be moved two cells to the right to obtain
<hexboard size="4x4"
coords="hide"
contents="R b1 E *:d1 E +:a2 E +:a3 E +:a4"
/>
respectively
<hexboard size="6x4"
coords="hide"
contents="E *:a1 E *:b1 R c1 E *:a2 R b2 E +:a4 E +:a5 E +:a6"
/>
.

Talk:Theory of ladder escapes

2016-05-29T14:25:21Z

Wurfmaul: /* About third row ladder escapes */ new section

== About third row ladder escapes ==

By definition, a third row ladder escape is a pattern P, for that holds: A+n+P is a template for all n≥0. This equivalent to: A+P and A+1+P are templates.

'''Proof.''' That the condition is necessary is obvious. That it is sufficient follows by induction from the following: For all n≥0, If A+n+P and A+(n+1)+P are templates, then A+(n+2)+P is a template, too. This is true because depending on where blue plays, red can reduce A+(n+2)+P to either A+n+P or A+(n+1)+P. For example, n=0:
<hexboard size="3x6"
coords="show"
contents="B a1 R b1 E *:e1 E *:f1 B a2 E *:e2 E *:f2 E *:e3 E *:f3"
/>
If blue plays anywhere except b2, a3 or b3, then red plays b2 and connects immediately. If blue plays b2, then red plays c1 and reduces the position to A+(n+1)+P. If blue plays a3 or b3, then red first plays at b2, forcing blue to play b3 or a3. Afterwards red plays d1 and reduces the position to A+n+P. ∎

In practice, in showing that A+1+P is a template, one may assume that blue drops to the first line, because otherwise red can reduce to A+P. This means: 
P is a third row ladder escape if and only if A+P and C+P are templates, where A and C are the patterns
<hexboard size="3x2"
coords="hide"
contents="E *:a1 R b1 E *:a2"
/>
and
<hexboard size="3x2"
coords="hide"
contents="R a1 a2"
/>
. 
Perhaps there is an equivalent proposition which requires even fewer amount of work to decide, I don't know.

For the same reason, i.e. that red may raise the ladder at will, if a pattern P is a third row ladder escape, then D+P is a second row ladder escape, where D is
<hexboard size="3x4"
coords="hide"
contents="E *:a1 +:a2 +:a3 +:d1 +:d2 +:d3"
/>
. 
That means every third row ladder escape is a second row ladder escape, if we move the beginning of the ladder two cells to the left, and assume that these 6 cells in D are empty. The <strike>ladder</strike> latter is natural to assume, since we previously expected a third ladder to come from there.
For example, let P be the following pattern:
<hexboard size="4x4"
coords="hide"
contents="E +:a2 +:a3 +:a4 *:d1 R c1"
/>
With the proposition from above, one can see that this is a third row ladder escape. Therefore, D+P, that is
<hexboard size="4x6"
coords="hide"
contents="E *:a1 *:a2 *:b1 +:a3 +:a4 *:f1 R e1"
/>
is a second row ladder escape. The latter is the same as your last example from the section "2nd row ladder escapes", except with an unnecessary cell on the fourth row.

At the same time,
<hexboard size="4x4"
coords="hide"
contents="E +:a3 +:a4 *:d1 R c1"
/>
and
<hexboard size="4x5"
coords="hide"
contents="E +:a3 +:a4 *:e1 R d1"
/>
are no second row ladder escapes, so in this case it is really necessary to move the start of the ladder by two cells.

--[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 14:25, 29 May 2016 (UTC)

Talk:Edge template V1b

2016-05-29T11:02:06Z

Wurfmaul:

Re: "(Note: As I am writing this I have only seen the claim of this being a valid template on the little golem thread. I have not checked it yet and also not if it is minimal. However, as this came from a very competent player I have no reson to doubt it.)"

This comment seems to cast some suspicion on whether the template is a template. It only takes benzene (computer program) around 10 seconds to check that this is definitely a template so I just wanted to stress that in my mind there is no doubt at all.

--wccanard

Re: e3 defence (currently not written) -- benzene (who might be playing a bit randomly as white) says that one main line is

e11 f10 f11 g10 g11 d11 c13 h10 h11 i10 i11 d12 d13 e12 e13 f12 f13 g12 g13 h12 h13 m12 j10

[Take away 8 from all the numbers (and leave the letters alone) to see the moves using the convention you're using (I'm on a standard 14x14 board so bottom row is 14 not 6)]

-- wccanard

I brought attention to this template on May 6th in [https://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=662 this littlegolem thread], before shalev reminded wccanard of it on May 19th in the thread that's linked in the article. Shalev had apparently known this template before, as have several others, I guess. It's well possible that someone knew about it before I was born. It's ok for me if the article stays as it is, I just have to mention it. I was a little disappointed to see that "a very competent player" didn't refer to me :-) --[[User:Wurfmaul|Wurfmaul]] ([[User talk:Wurfmaul|talk]]) 11:02, 29 May 2016 (UTC)

Online playing

2016-05-25T21:44:10Z

Wurfmaul: /* Services for game analysis */

Hex can be played online either using a web-based format or via e-mail.

=Web-based games=

Web-based games can be played either in ''real-time'', where moves are made within minutes (or even seconds), or ''turn-based'', where a player has days for one move.

==Realtime playing sites==

The most popular sites (approximatively ordered by the number of hex games played on them daily) are:
* [[igGameCenter]] for real-time play, with time settings, and ranking (it has many connection games).
* [http://en.boardgamearena.com Board Game Arena] supports real-time as well as turn-based play
* [[boardspace]] for real-time play
* http://www.ludoteka.com/ for real-time play
* http://games.wtanaka.com/hex for real-time or turn-based play

==Turn-based playing sites==

* [[Little Golem]] for turn-based play (also has other games).
* [http://en.boardgamearena.com Board Game Arena]
* See also [http://www.gamerz.net/pbmserv Richard's server]. One can play completely by e-mail, but it also has a [http://www.gamerz.net/pbmserv/List.php?Hex graphical interface] now. Furthermore any sized board is supported.

=E-mail-based games=

Hex may also be played over e-mail, in a turn-based fashion. The board can be represented in ASCII using either the full or compact formats below. The full layout is rotated 90 degrees from the compact one. (A fixed-width font is required for either board to display correctly in an e-mail client.)

<pre>
Full layout:

O _ X
O _/ \_ X
O _/ \_/ \_ X
O _/ \_/ \_/ \_ X
O _/ \_/ \_/ \_/ \_ X
O _/ \_/ \_/ \_/ \_/ \_ X
O _/ \_/ \_/ \_/ \_/ \_/ \_ X
O _/ \_/ \_/ \_/ \_/ \_/ \_/ \_ X
_/ \_/ \_/O\_/ \_/ \_/ \_/ \_/ \_
/ \_/ \_/ \_/ \_/X\_/ \_/ \_/ \_/ \
\_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/ \_/
X\_/ \_/ \_/O\_/O\_/X\_/ \_/ \_/O
X\_/ \_/ \_/ \_/ \_/ \_/ \_/O
X\_/ \_/ \_/X\_/ \_/ \_/O
X\_/ \_/ \_/ \_/ \_/O
X\_/ \_/ \_/ \_/O
X\_/ \_/ \_/O
X\_/ \_/O
X\_/O

Compact layout:
/.\
/. .\
/. . .\
X /. . . .\ O
/. . . . .\
/. . . O . .\
/. . O . . . .\
/. . . . . . . .\
. . X O X . . . .
\. . . . . . . ./
\. . X . . . ./
\. . . . . ./
\. . . . ./
O \. . . ./ X
\. . ./
\. ./
\./
</pre>

==Services for game analysis==

There are some services in the net, which help one play out different variations, analyze the games and share game records.

* [http://www.trmph.com/ TRMPH] has 3 board sizes: 11, 13 and 19.

* [http://www.hexmaster.net/ Hex master] is a service for analyzing and commenting games from Little golem.

* [http://gwylim.net/hexeditor Hex editor] (dead link) by Gwylim Ashley. Still in development. Support any size of the board, and creates different lines of variations, labeled with numbers.

All 3 websites can import games from Little golem.

[[category: Hex community]]
[[category: online play]]

Parallelogram boards

2016-04-07T14:05:39Z

Wurfmaul: Created page with "Hex is usually played on a rhombic n×n board, but one can also try playing it on n×m parallelogram boards, where n is the number of rows, m the number of columns, and n ≠..."

Hex is usually played on a rhombic n×n board, but one can also try playing it on n×m parallelogram boards, where n is the number of rows, m the number of columns, and n ≠ m. However, there is a simple [[symmetry winning strategy]] for the player with the shorter distance between his sides, even when he moves second. To mitigate this, one can allow the player with the greater distance between his sides to begin the game and place a certain number of pieces at once in her first move. In particular, it has been found that Hex on a 7×9 board is a rather fair game, when the vertical player may start the game with two pieces at once.

Number of pieces head start the vertical player needs to force a win:
{| class="wikitable" style="text-align: center;"
! ×
! scope="column" | 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12
|-
! scope="row" | 1
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12
|-
! scope="row" | 2
| 0 || 1 ||2 || 2 || 3 || 4 || 4 || 5 || 6 || 6 || 7 || 8
|-
! scope="row" | 3
| 0 || 0 || 1 || 2 || 3 || 3 || 4 || (≤)5 || || || ||
|-
! scope="row" | 4
| 0 || 0 || 0 || 1 || 2 || 2 || 3 || (≤)4 || || || ||
|-
! scope="row" | 5
| 0 || 0 || 0 || 0 || 1 || 2 || 2 || || || || (≤)5 ||
|-
! scope="row" | 6
| 0 || 0 || 0 || 0 || 0 || 1 || 2 || 2 || || || || (≤)5
|-
! scope="row" | 7
| 0 || 0 || 0 || 0 || 0 || 0 || 1 || 2 || 2 || || ||
|-
! scope="row" | 8
| 0 || 0 || 0 || 0 || 0 || 0 || 0 || 1 || 2 || 2? || ||
|}

User:Wurfmaul

2016-04-07T13:35:39Z

Wurfmaul:

[[Parallelogram boards]]

Small boards

2016-04-04T10:47:41Z

Wurfmaul: /* Size 8 */

Playing [[Hex]] on [[board]]s of size smaller than 10 × 10 is not very interesting, since many players will be able to play almost perfectly. However it may still be intersting for theoretical studies, and for making [[Puzzles|problems]].

The boards of size up to five can be solved by hand. Hex on 6 × 6 has been solved by [[Queenbee]].

Here are the winning first moves on the small boards. [[Red (player)|Red]] is vertical and plays first. The [[Hex (board element)|cells]] containing a red [[Piece|stone]] are winning moves for red, while those containing a blue stone are losing. For more details, visit Queenbee's own [http://www.cs.ualberta.ca/~queenbee/openings.html opening page].

''Update:'' The 7 × 7 board has been solved by [[Ryan Hayward|R. Hayward]], et.al. For more details, visit http://www.cs.ualberta.ca/~hayward/hex7trees/

== Winner depending on the first move ==
The following boards can help you decide where you should [[swap]] when playing on small boards, and it might give you ideas of patterns for bigger boards.
<hex>R2 C2 Q1 Vb1 Va2 Ha1 Hb2</hex>

<hex>R3 C3 Q1 Va2 Va3 Hb1 Vb2 Hb3 Vc1 Vc2 Ha1 Hc3</hex>

<hex>R4 C4 Q1 Va4 Vb3 Vc2 Vd1 Ha1 Ha2 Ha3 Hb1 Hb2 Hb4 Hc1 Hc3 Hc4 Hd2 Hd3 Hd4</hex>

<hex>R5 C5 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Ha3 Hb1 Hc1 Hd1 Hb5 Hc5 Hd5 He5 He4 He3</hex>

<hex>R6 C6 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Va3 Hb1 Hc1 Hd1 He1 Vb5 Vc5 Vd5 Ve5 Ve4 Ve3 Vf1 Vf2 Vf3 Vf4 Hf5 Hf6 He6 Hd6 Hc6 Hb6 Va6</hex>

=== Size 7 ===

Size 7 was first solved by [[Ryan Hayward]] using [[domination]].

<hex>R7 C7 Q1 Ha1 Hb1 Hc1 Hd1 He1 Hf1 Vg1 Ha2 Hb2 Vc2 Hd2 Ve2 Vf2 Vg2 Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Hg3 Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Ha5 Vb5 Vc5 Vd5 Ve5 Vf5 Hg5 Va6 Vb6 Vc6 Hd6 Ve6 Hf6 Hg6 Va7 Hb7 Hc7 Hd7 He7 Hf7 Hg7</hex>

=== Size 8 ===

The outcomes for size 8 were computer generated by [[Javerberg]]. The solution was independently computer generated by Hayward et al. and appeared in [[INJCAI|IJCAI09]].

<hex>
R8 C8 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Vh1
Ha2 Hb2 Hc2 Hd2 He2 Hf2 Vg2 Vh2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Hh3
Ha4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Hh5
Ha6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Hh6
Va7 Vb7 Hc7 Hd7 He7 Hf7 Hg7 Hh7
Va8 Hb8 Hc8 Hd8 He8 Hf8 Hg8 Hh8
</hex>

=== Size 9 ===

The outcomes for size 9 are computer generated by [[University of Alberta]]'s Hex group.

<hex>
R9 C9 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Hh1 Vi1
Va2 Vb2 Vc2 Hd2 He2 Hf2 Hg2 Vh2 Vi2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Vh3 Hi3
Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4 Vi4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Vh5 Vi5
Va6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Vh6 Vi6
Ha7 Vb7 Vc7 Vd7 Ve7 Vf7 Vg7 Vh7 Hi7
Va8 Vb8 Hc8 Hd8 He8 Hf8 Vg8 Vh8 Vi8
Va9 Hb9 Hc9 Hd9 He9 Hf9 Hg9 Hh9 Hi9
</hex>

== Reference ==
* [[Queenbee]]'s opening [http://www.cs.ualberta.ca/~queenbee/openings.html page] is a reference for sizes under 6x6.
* This [http://www.ru.is/faculty/yngvi/pdf/HaywardBJKPR05.pdf article] by Ryan Hayward ''et al.'' is a reference for 7x7.
* This [[Little Golem]]'s forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=338 thread] is a reference for size 8x8.
* This [https://webdocs.cs.ualberta.ca/~hayward/papers/pawlhayw.pdf article] by Jakub Pawlewicz and Ryan Hayward is a reference for size 9x9.

== See also ==

* [[Board size]]
* [[Jing Yang]] designed a [[decomposition method]] to find winning strategy in Hex. [http://www.ee.umanitoba.ca/~jingyang/index.html Home Page].

[[Category: Theory]]

Small boards

2016-04-04T10:45:17Z

Wurfmaul: /* Reference */ 9x9 -> size 9x9

Playing [[Hex]] on [[board]]s of size smaller than 10 × 10 is not very interesting, since many players will be able to play almost perfectly. However it may still be intersting for theoretical studies, and for making [[Puzzles|problems]].

The boards of size up to five can be solved by hand. Hex on 6 × 6 has been solved by [[Queenbee]].

Here are the winning first moves on the small boards. [[Red (player)|Red]] is vertical and plays first. The [[Hex (board element)|cells]] containing a red [[Piece|stone]] are winning moves for red, while those containing a blue stone are losing. For more details, visit Queenbee's own [http://www.cs.ualberta.ca/~queenbee/openings.html opening page].

''Update:'' The 7 × 7 board has been solved by [[Ryan Hayward|R. Hayward]], et.al. For more details, visit http://www.cs.ualberta.ca/~hayward/hex7trees/

== Winner depending on the first move ==
The following boards can help you decide where you should [[swap]] when playing on small boards, and it might give you ideas of patterns for bigger boards.
<hex>R2 C2 Q1 Vb1 Va2 Ha1 Hb2</hex>

<hex>R3 C3 Q1 Va2 Va3 Hb1 Vb2 Hb3 Vc1 Vc2 Ha1 Hc3</hex>

<hex>R4 C4 Q1 Va4 Vb3 Vc2 Vd1 Ha1 Ha2 Ha3 Hb1 Hb2 Hb4 Hc1 Hc3 Hc4 Hd2 Hd3 Hd4</hex>

<hex>R5 C5 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Ha3 Hb1 Hc1 Hd1 Hb5 Hc5 Hd5 He5 He4 He3</hex>

<hex>R6 C6 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Va3 Hb1 Hc1 Hd1 He1 Vb5 Vc5 Vd5 Ve5 Ve4 Ve3 Vf1 Vf2 Vf3 Vf4 Hf5 Hf6 He6 Hd6 Hc6 Hb6 Va6</hex>

=== Size 7 ===

Size 7 was first solved by [[Ryan Hayward]] using [[domination]].

<hex>R7 C7 Q1 Ha1 Hb1 Hc1 Hd1 He1 Hf1 Vg1 Ha2 Hb2 Vc2 Hd2 Ve2 Vf2 Vg2 Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Hg3 Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Ha5 Vb5 Vc5 Vd5 Ve5 Vf5 Hg5 Va6 Vb6 Vc6 Hd6 Ve6 Hf6 Hg6 Va7 Hb7 Hc7 Hd7 He7 Hf7 Hg7</hex>

=== Size 8 ===

The outcomes for size 8 were computer generated by [[Javerberg]]. The solution was independantly computer generated by Hayward et al. and appeared in [[INJCAI|IJCAI09]].

<hex>
R8 C8 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Vh1
Ha2 Hb2 Hc2 Hd2 He2 Hf2 Vg2 Vh2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Hh3
Ha4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Hh5
Ha6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Hh6
Va7 Vb7 Hc7 Hd7 He7 Hf7 Hg7 Hh7
Va8 Hb8 Hc8 Hd8 He8 Hf8 Hg8 Hh8
</hex>

=== Size 9 ===

The outcomes for size 9 are computer generated by [[University of Alberta]]'s Hex group.

<hex>
R9 C9 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Hh1 Vi1
Va2 Vb2 Vc2 Hd2 He2 Hf2 Hg2 Vh2 Vi2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Vh3 Hi3
Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4 Vi4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Vh5 Vi5
Va6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Vh6 Vi6
Ha7 Vb7 Vc7 Vd7 Ve7 Vf7 Vg7 Vh7 Hi7
Va8 Vb8 Hc8 Hd8 He8 Hf8 Vg8 Vh8 Vi8
Va9 Hb9 Hc9 Hd9 He9 Hf9 Hg9 Hh9 Hi9
</hex>

== Reference ==
* [[Queenbee]]'s opening [http://www.cs.ualberta.ca/~queenbee/openings.html page] is a reference for sizes under 6x6.
* This [http://www.ru.is/faculty/yngvi/pdf/HaywardBJKPR05.pdf article] by Ryan Hayward ''et al.'' is a reference for 7x7.
* This [[Little Golem]]'s forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=338 thread] is a reference for size 8x8.
* This [https://webdocs.cs.ualberta.ca/~hayward/papers/pawlhayw.pdf article] by Jakub Pawlewicz and Ryan Hayward is a reference for size 9x9.

== See also ==

* [[Board size]]
* [[Jing Yang]] designed a [[decomposition method]] to find winning strategy in Hex. [http://www.ee.umanitoba.ca/~jingyang/index.html Home Page].

[[Category: Theory]]

User:Wurfmaul

2016-04-04T10:00:33Z

Wurfmaul: Blanked the page

Deleted pages

2016-04-04T09:40:37Z

Wurfmaul:

I decided that creating this page was not a good idea after all.
I wasn't aware that I couldn't delete the page when I created it. Please delete it (and also the redirect [[Discussion_of_7x9_Hex]] and the subpages [[Discussion_of_7x9_Hex/0]] and [[Discussion_of_7x9_Hex/1]] if you can!

Deleted pages

2016-04-04T09:35:32Z

Wurfmaul:

I decided that creating this page was not a good idea after all.
I wasn't aware that I couldn't delete the page when I created it. Please delete it if you can!

Deleted pages

2016-04-04T09:30:57Z

Wurfmaul: Wurfmaul moved page Discussion of 7x9 Hex to Wurfmaul/trash/Discussion of 7x9 Hex: I just decided it was a bad idea to create the page and it seems I can't delete it.

This page has been created in response to the 7x9 discussion here: [https://littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=649]

7x9 is far too difficult for [[user:Wurfmaul|me]] on my own. All this analysis has been done with the benzene dfpn commands.

Red is the vertical player, blue is the horizontal player. (This seems to be standard on hexwiki?) Blue begins the game by playing two moves in a row.

[[Discussion of 7x9 Hex/0|Can blue win?]]

Arek suggests the following:
<hexboard
size="7x9"
contents="B d4 f5"
/>
[[Discussion of 7x9 Hex/1|Does it work?]]

Deleted pages

2016-04-04T09:15:00Z

Wurfmaul:

Deleted pages

2016-04-04T09:14:04Z

Wurfmaul:

This page has been created in response to the 7x9 discussion here: [https://littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=649]

7x9 is far too difficult for [[user:Wurfmaul|me]] on my own. All this analysis has been done with the benzene dfpn commands.

Red is the vertical player, blue is the horizontal player. (This seems to be standard on hexwiki?) Blue begins the game by playing two moves in a row.

Arek suggests the following:
<hexboard
size="7x9"
contents="B d4 f5"
/>
[[Discussion of 7x9 Hex/1|Does it work?]]

Deleted pages

2016-04-04T09:08:06Z

Wurfmaul:

This page has been created in response to the 7x9 discussion here: [https://littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=649]

7x9 is far too difficult for [[Wurfmaul|me]] on my own. All this analysis has been done with the benzene dfpn commands.

Red is the vertical player, blue is the horizontal player. (This seems to be standard on hexwiki?) Blue begins the game by playing two moves in a row.

Arek suggests the following:
<hexboard
size="7x9"
contents="B d4 f5"
/>
[[Discussion of 7x9 Hex/1|Does it work?]]

Deleted pages

2016-04-04T08:58:27Z

Wurfmaul: Created page with "This page has been created in response to the 7x9 discussion here: [https://littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=649] Red is the vertical player, blue is the ho..."

This page has been created in response to the 7x9 discussion here: [https://littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=649]
Red is the vertical player, blue is the horizontal player (this seems to be standard on hexwiki?)
Blue begins the game by playing two moves in a row.

Arek suggests the following:
<hexboard
size="7x9"
contents="B d4 f5"
/>
[[Discussion of 7x9 Hex/1|Does it work?]]

User:Wurfmaul

2016-04-04T08:36:49Z

Wurfmaul:

[[Discussion of 7x9 Hex]]

User:Wurfmaul

2016-04-04T07:12:45Z

Wurfmaul: Blanked the page

User:Wurfmaul

2016-04-04T07:11:40Z

Wurfmaul: test test

<spoiler show="show_message">spoiler_text</spoiler>

Small boards

2016-04-04T05:42:41Z

Wurfmaul: /* Reference */ reference for 9x9

Playing [[Hex]] on [[board]]s of size smaller than 10 × 10 is not very interesting, since many players will be able to play almost perfectly. However it may still be intersting for theoretical studies, and for making [[Puzzles|problems]].

The boards of size up to five can be solved by hand. Hex on 6 × 6 has been solved by [[Queenbee]].

Here are the winning first moves on the small boards. [[Red (player)|Red]] is vertical and plays first. The [[Hex (board element)|cells]] containing a red [[Piece|stone]] are winning moves for red, while those containing a blue stone are losing. For more details, visit Queenbee's own [http://www.cs.ualberta.ca/~queenbee/openings.html opening page].

''Update:'' The 7 × 7 board has been solved by [[Ryan Hayward|R. Hayward]], et.al. For more details, visit http://www.cs.ualberta.ca/~hayward/hex7trees/

== Winner depending on the first move ==
The following boards can help you decide where you should [[swap]] when playing on small boards, and it might give you ideas of patterns for bigger boards.
<hex>R2 C2 Q1 Vb1 Va2 Ha1 Hb2</hex>

<hex>R3 C3 Q1 Va2 Va3 Hb1 Vb2 Hb3 Vc1 Vc2 Ha1 Hc3</hex>

<hex>R4 C4 Q1 Va4 Vb3 Vc2 Vd1 Ha1 Ha2 Ha3 Hb1 Hb2 Hb4 Hc1 Hc3 Hc4 Hd2 Hd3 Hd4</hex>

<hex>R5 C5 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Ha3 Hb1 Hc1 Hd1 Hb5 Hc5 Hd5 He5 He4 He3</hex>

<hex>R6 C6 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Va3 Hb1 Hc1 Hd1 He1 Vb5 Vc5 Vd5 Ve5 Ve4 Ve3 Vf1 Vf2 Vf3 Vf4 Hf5 Hf6 He6 Hd6 Hc6 Hb6 Va6</hex>

=== Size 7 ===

Size 7 was first solved by [[Ryan Hayward]] using [[domination]].

<hex>R7 C7 Q1 Ha1 Hb1 Hc1 Hd1 He1 Hf1 Vg1 Ha2 Hb2 Vc2 Hd2 Ve2 Vf2 Vg2 Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Hg3 Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Ha5 Vb5 Vc5 Vd5 Ve5 Vf5 Hg5 Va6 Vb6 Vc6 Hd6 Ve6 Hf6 Hg6 Va7 Hb7 Hc7 Hd7 He7 Hf7 Hg7</hex>

=== Size 8 ===

The outcomes for size 8 were computer generated by [[Javerberg]]. The solution was independantly computer generated by Hayward et al. and appeared in [[INJCAI|IJCAI09]].

<hex>
R8 C8 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Vh1
Ha2 Hb2 Hc2 Hd2 He2 Hf2 Vg2 Vh2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Hh3
Ha4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Hh5
Ha6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Hh6
Va7 Vb7 Hc7 Hd7 He7 Hf7 Hg7 Hh7
Va8 Hb8 Hc8 Hd8 He8 Hf8 Hg8 Hh8
</hex>

=== Size 9 ===

The outcomes for size 9 are computer generated by [[University of Alberta]]'s Hex group.

<hex>
R9 C9 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Hh1 Vi1
Va2 Vb2 Vc2 Hd2 He2 Hf2 Hg2 Vh2 Vi2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Vh3 Hi3
Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4 Vi4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Vh5 Vi5
Va6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Vh6 Vi6
Ha7 Vb7 Vc7 Vd7 Ve7 Vf7 Vg7 Vh7 Hi7
Va8 Vb8 Hc8 Hd8 He8 Hf8 Vg8 Vh8 Vi8
Va9 Hb9 Hc9 Hd9 He9 Hf9 Hg9 Hh9 Hi9
</hex>

== Reference ==
* [[Queenbee]]'s opening [http://www.cs.ualberta.ca/~queenbee/openings.html page] is a reference for sizes under 6x6.
* This [http://www.ru.is/faculty/yngvi/pdf/HaywardBJKPR05.pdf article] by Ryan Hayward ''et al.'' is a reference for 7x7.
* This [[Little Golem]]'s forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=338 thread] is a reference for size 8x8.
* This [https://webdocs.cs.ualberta.ca/~hayward/papers/pawlhayw.pdf article] by Jakub Pawlewicz and Ryan Hayward is a reference for 9x9.

== See also ==

* [[Board size]]
* [[Jing Yang]] designed a [[decomposition method]] to find winning strategy in Hex. [http://www.ee.umanitoba.ca/~jingyang/index.html Home Page].

[[Category: Theory]]

Small boards

2016-04-04T05:40:28Z

Wurfmaul: /* Size 9 */ added the remaining opening moves

Playing [[Hex]] on [[board]]s of size smaller than 10 × 10 is not very interesting, since many players will be able to play almost perfectly. However it may still be intersting for theoretical studies, and for making [[Puzzles|problems]].

The boards of size up to five can be solved by hand. Hex on 6 × 6 has been solved by [[Queenbee]].

Here are the winning first moves on the small boards. [[Red (player)|Red]] is vertical and plays first. The [[Hex (board element)|cells]] containing a red [[Piece|stone]] are winning moves for red, while those containing a blue stone are losing. For more details, visit Queenbee's own [http://www.cs.ualberta.ca/~queenbee/openings.html opening page].

''Update:'' The 7 × 7 board has been solved by [[Ryan Hayward|R. Hayward]], et.al. For more details, visit http://www.cs.ualberta.ca/~hayward/hex7trees/

== Winner depending on the first move ==
The following boards can help you decide where you should [[swap]] when playing on small boards, and it might give you ideas of patterns for bigger boards.
<hex>R2 C2 Q1 Vb1 Va2 Ha1 Hb2</hex>

<hex>R3 C3 Q1 Va2 Va3 Hb1 Vb2 Hb3 Vc1 Vc2 Ha1 Hc3</hex>

<hex>R4 C4 Q1 Va4 Vb3 Vc2 Vd1 Ha1 Ha2 Ha3 Hb1 Hb2 Hb4 Hc1 Hc3 Hc4 Hd2 Hd3 Hd4</hex>

<hex>R5 C5 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Ha3 Hb1 Hc1 Hd1 Hb5 Hc5 Hd5 He5 He4 He3</hex>

<hex>R6 C6 Q1 Ve1 Vb2 Vc2 Vd2 Ve2 Vb3 Vc3 Vd3 Va4 Vb4 Vc4 Vd4 Va5 Ha1 Ha2 Va3 Hb1 Hc1 Hd1 He1 Vb5 Vc5 Vd5 Ve5 Ve4 Ve3 Vf1 Vf2 Vf3 Vf4 Hf5 Hf6 He6 Hd6 Hc6 Hb6 Va6</hex>

=== Size 7 ===

Size 7 was first solved by [[Ryan Hayward]] using [[domination]].

<hex>R7 C7 Q1 Ha1 Hb1 Hc1 Hd1 He1 Hf1 Vg1 Ha2 Hb2 Vc2 Hd2 Ve2 Vf2 Vg2 Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Hg3 Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Ha5 Vb5 Vc5 Vd5 Ve5 Vf5 Hg5 Va6 Vb6 Vc6 Hd6 Ve6 Hf6 Hg6 Va7 Hb7 Hc7 Hd7 He7 Hf7 Hg7</hex>

=== Size 8 ===

The outcomes for size 8 were computer generated by [[Javerberg]]. The solution was independantly computer generated by Hayward et al. and appeared in [[INJCAI|IJCAI09]].

<hex>
R8 C8 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Vh1
Ha2 Hb2 Hc2 Hd2 He2 Hf2 Vg2 Vh2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Hh3
Ha4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Hh5
Ha6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Hh6
Va7 Vb7 Hc7 Hd7 He7 Hf7 Hg7 Hh7
Va8 Hb8 Hc8 Hd8 He8 Hf8 Hg8 Hh8
</hex>

=== Size 9 ===

The outcomes for size 9 are computer generated by [[University of Alberta]]'s Hex group.

<hex>
R9 C9 Q1
Ha1 Hb1 Hc1 Hd1 He1 Hf1 Hg1 Hh1 Vi1
Va2 Vb2 Vc2 Hd2 He2 Hf2 Hg2 Vh2 Vi2
Ha3 Vb3 Vc3 Vd3 Ve3 Vf3 Vg3 Vh3 Hi3
Va4 Vb4 Vc4 Vd4 Ve4 Vf4 Vg4 Vh4 Vi4
Va5 Vb5 Vc5 Vd5 Ve5 Vf5 Vg5 Vh5 Vi5
Va6 Vb6 Vc6 Vd6 Ve6 Vf6 Vg6 Vh6 Vi6
Ha7 Vb7 Vc7 Vd7 Ve7 Vf7 Vg7 Vh7 Hi7
Va8 Vb8 Hc8 Hd8 He8 Hf8 Vg8 Vh8 Vi8
Va9 Hb9 Hc9 Hd9 He9 Hf9 Hg9 Hh9 Hi9
</hex>

== Reference ==
* [[Queenbee]]'s opening [http://www.cs.ualberta.ca/~queenbee/openings.html page] is a reference for sizes under 6x6.
* This [http://www.ru.is/faculty/yngvi/pdf/HaywardBJKPR05.pdf article] by Ryan Hayward ''et al.'' is a reference for 7x7.
* This [[Little Golem]]'s forum [http://www.littlegolem.net/jsp/forum/topic2.jsp?forum=50&topic=338 thread] is a reference for size 8x8.

== See also ==

* [[Board size]]
* [[Jing Yang]] designed a [[decomposition method]] to find winning strategy in Hex. [http://www.ee.umanitoba.ca/~jingyang/index.html Home Page].

[[Category: Theory]]