HexWiki - User contributions [en]

User:Alcalyn

2025-06-17T07:18:12Z

Alcalyn: Replaced content with "Hello!"

Hello!

User:Alcalyn

2025-06-17T07:14:36Z

Alcalyn:

Hello!

<hexboard size="10x10000"
coords="hide"
contents="R f2 R g2 R h2 R h2 R i2 R e3 R i3 R d4 E *:f4 E *:h4 R i4 R c5 R i5 R b6 B d6 B h6 R i6 R b7 B d7 B e7 B f7 B g7 R h7 R b8 R h8 R b9 R c9 R d9 R e9 R f9 R g9"
/>

User:Alcalyn

2025-06-17T07:11:25Z

Alcalyn:

Hello!

<hexboard size="10x10000"
coords="hide"
contents="R e2 R f2 R g2 R h2 R i2 R d3 R i3 R c4 E *:e4 E *:g4 R i4 R b5 R i5 R a6 B c6 B g6 R i6 R a7 B c7 B d7 B e7 B f7 R h7 R a8 R g8 R a9 R b9 R c9 R d9 R e9 R f9"
/>

User:Alcalyn

2025-06-17T07:11:04Z

Alcalyn:

Hello!

<hexboard size="10x10"
coords="show"
contents="R e2 R f2 R g2 R h2 R i2 R d3 R i3 R c4 E *:e4 E *:g4 R i4 R b5 R i5 R a6 B c6 B g6 R i6 R a7 B c7 B d7 B e7 B f7 R h7 R a8 R g8 R a9 R b9 R c9 R d9 R e9 R f9"
/>

User:Alcalyn

2025-06-17T07:10:23Z

Alcalyn:

Hello!

<hexboard size="9x9"
coords="hide"
contents="R e2 R f2 R g2 R h2 R i2 R d3 R i3 R c4 E *:e4 E *:g4 R i4 R b5 R i5 R a6 B c6 B g6 R i6 R a7 B c7 B d7 B e7 B f7 R h7 R a8 R g8 R a9 R b9 R c9 R d9 R e9 R f9"
/>

User:Alcalyn

2025-06-17T07:08:42Z

Alcalyn:

Hello!

<hexboard size="6x4"
coords="show"
contents="B 5:b1 B 5:c2 R 6:a3 R 4:d3 R 2:b4 R 2:b5"
/>

PlayHex

2025-02-11T08:59:15Z

Alcalyn: github repo moved to an organization

PlayHex is a website for playing Hex online. It supports all board sizes from 1x1 to 42x42, and flexible time controls for both real-time and turn-based play. It also provides AI opponents of various strengths, including a very strong version of [[KataHex]]. Moreover, the website offers almost instantaneous AI-based game analysis.

== Open source development ==

The PlayHex project, including the Hex server and client, is free software under an open source license. Its creator and principal developer is [[User:Alcalyn|Alcalyn]], with additional contributions by [[User:Comonoid|Comonoid]] and others. The source code is hosted on [https://github.com/playhex/playhex GitHub].

== History ==

Development started in December 2022 and the playhex.org domain was registered at the same time, but independently, by [[User:Mason|Mason]]. On December 18, 2023, an early version of the still unnamed PlayHex website was made available to the community on the [[Hex_forums|Hex Discord Forum]]. The site went live at playhex.org in February 2024. As of May 2024, the project continues to be under active development.

== External links ==

* [https://playhex.org playhex.org]
* [https://github.com/playhex/playhex GitHub repository for PlayHex]

[[category: Online play]]
[[category: Hex community]]

PlayHex

2025-02-11T08:58:31Z

Alcalyn: /* External links */ github repo moved to an organization

PlayHex is a website for playing Hex online. It supports all board sizes from 1x1 to 42x42, and flexible time controls for both real-time and turn-based play. It also provides AI opponents of various strengths, including a very strong version of [[KataHex]]. Moreover, the website offers almost instantaneous AI-based game analysis.

== Open source development ==

The PlayHex project, including the Hex server and client, is free software under an open source license. Its creator and principal developer is [[User:Alcalyn|Alcalyn]], with additional contributions by [[User:Comonoid|Comonoid]] and others. The source code is hosted on [https://github.com/alcalyn/hex/ GitHub].

== History ==

Development started in December 2022 and the playhex.org domain was registered at the same time, but independently, by [[User:Mason|Mason]]. On December 18, 2023, an early version of the still unnamed PlayHex website was made available to the community on the [[Hex_forums|Hex Discord Forum]]. The site went live at playhex.org in February 2024. As of May 2024, the project continues to be under active development.

== External links ==

* [https://playhex.org playhex.org]
* [https://github.com/playhex/playhex GitHub repository for PlayHex]

[[category: Online play]]
[[category: Hex community]]

Conventions

2025-01-24T20:07:31Z

Alcalyn: PlayHex does not have negative orientations

This page lists possible conventions for Hex games. These conventions concern the color of the players, who goes first, the orientation of the board, and the numbering of the positions on the board. Unlike much older games such as Chess and Go, there are no universally agreed upon conventions for Hex, and different people, game sites, books, and computer programs sometimes use different conventions.

We distinguish ''logical'' from ''physical'' conventions. Logical conventions are concerned with the abstract rules of Hex, irrespectively of how the players visualize the game. Physical conventions relate to the specific appearance of the game board. When the game is played remotely, for example on an internet game server or over email, it is in principle possible for the two players to follow different physical conventions; for example, each player may choose their own preferred orientation of the board, or their own preferred colors for the pieces. However, both players should follow the same logical conventions.

== Universal vs. local conventions ==

There are several aspects of a game of Hex that are arbitrary and do not affect game play in an essential way. This includes the orientation (rotation and reflection) of the board, the colors of the players, who goes first, which edge belongs to which player, and whether and how the cells on the board are numbered.

The same also applies to other games; for example, the game of Chess would not change in an essential way if the white square were in the bottom left corner instead of the bottom right one, if black went first instead of white, if the white queen started on a black square instead of a white one, if the ranks were lettered and the files numbered instead of the other way around, and so on.

In the case of established games such as Chess and Go, there are universal conventions that all players have agreed on. This has certain advantages. For example, every chess player understands what it means to move a white pawn from d2 to d4, without requiring further explanation.

In Hex, there were no universal conventions prior to ca. 2020, and different players, books, game sites, tournaments, or Hex programs often used different conventions. Fortunately, some universal conventions seem to have crystallized in recent years, especially for the logical aspects of the game (such as the coordinate system and who goes first). Some physical conventions also seem to have become standard; for example, the colors are now almost universally black/white or red/blue, with black or red going first. Some conventions are unsettled; for example, both swap methods (swap-pieces and swap-sides) seem to be in common use. This is not a problem as long as game records specify which method is used. Other conventions, such as the board orientation, seem to depend on user preference and probably don't need to be standardized, although there is a strong preference for positive orientations.

== Logical conventions ==

=== Cell naming ===

Cells are arranged in a grid, with each cell named by a letter and a number. Letters can be uppercase or lowercase. The following convention seems to be nearly universally used:

[[Image:Logical-Board.png|right|]]

'''Convention: coordinate origin'''

Possible values:

* '''Acute:''' The cell A1 is an an acute corner of the board.

In particular, this means that the three cells A1, B1, and A2 are all adjacent to each other.

For the purpose of the following exposition, it is useful to refer to the cells that share a common number as a "rank", and to the cells that share a common letter as a "file", as in Chess. For example, the 1-rank consists of the cells A1, B1, C1, etc., and the A-file consists of the cells A1, A2, A3, etc.

Note that in some games, such as Go, it is customary to omit certain letters from the alphabet, especially the letter I. This is done "to avoid confusion between I and J", and presumably dates from a time when typesetting was uncommon and people had terrible handwriting. However, in Hex, the standard 26-letter alphabet is used. If more than 26 letters are needed, alphabet numbers can be used: the next file after Z is AA, then AB, and so on.

=== Edge coloring ===

To avoid referencing specific physical attributes of the game, we will refer to the color of the first piece played in the game as "color 1", and to the other color as "color 2". One pair of opposite edges "belongs to" color 1; these are the edges that the player who is playing color 1 is trying to connect. The other pair of opposite edges belongs to color 2.

'''Convention: edge coloring'''

Possible values:

* '''Normal:''' The edge that is adjacent to the 1-rank belongs to color 1, and the edge that is adjacent to the A-file belongs to color 2.

In other words, the normal edge coloring convention states that the color 1 edges are parallel to ranks, and the color 2 edges are parallel to files. In a typical rendering of the board, the letters (naming files) are written along the color 1 edge and the numbers (naming ranks) are written along the color 2 edge.

I am not aware of anybody who has used the opposite convention. However, there are some authors who use no edge coloring convention at all.

=== Swapping ===

There are two different ways of implementing the [[swap rule]]. Which convention is chosen will affect the notation for games. It is also possible to permit both methods of swapping; then it is up to the player to decide which method to use.

'''Convention: swap method'''

Possible values:

* '''Swap sides:''' Upon playing a swap move, the board position stays the same and the players change colors.

* '''Swap pieces:''' Upon playing a swap move, the players keep their colors and the board position is mirrored (ranks and files interchanged) and the color of the pieces is inverted. For example, a black piece at A2 would be replaced by a white piece at B1.

Here is an example using the swap sides convention: Player 1 plays a black piece at g4. Player 2 swaps sides. The board state remains unchanged, and immediately after the swap, it is player 1's turn to play a white piece.

Here is an example using the swap pieces convention: Player 1 plays a black piece at g4. Player 2 swaps pieces, and replaces the black piece at g4 with a white piece at d7. Immediately after the swap, it is player 1's turn to play a black piece.

When using algebraic notation for a sequence of moves, it is important to know which swap method was used. The swap method should either be defined for the context in which it is used, or else the notation should state explicitly which method was used. For example, with the swap sides convention, the notation "g4 swap f7 e7" means player 1 plays color 1 at g4, player 2 swaps, player 1 plays color 2 at f7, and player 2 plays color 1 at e7. With the swap pieces convention, the same game would be described as "g4 swap g6 g5". Here, player 1 plays color 1 at g4, player 2 replaces this by a piece by color 2 at d7, then player 1 plays color 1 at g6 and player 2 plays color 2 at g5. It is best not to use ambiguous notation at all; the two games can then be denoted "g4 swap-sides f7 e7" and "g4 swap-pieces g6 g5".

Note that the above logical conventions are purely symbolic. They do not depend on any particular board layout.

=== Passing ===

Although passing (skipping a move) is not always considered part of the classic Hex rules, allowing it does not change the nature of the game and has certain advantages. See the page on [[passing]] for more details.

'''Convention: passing'''

Possible values:

* '''Explicit''': Players can pass and there is an explicit passing move in the game record, e.g.: "Black passed".

* '''Implicit''': There is no explicit passing move, but a player can move twice in a row (presumably when the other player allows it).

* '''None''': Moves must be strictly alternating and passing is not allowed.

== Physical conventions ==

=== Colors ===

The most common color schemes are:

* Black and white.
* Red and blue.
* V and H.

With each color scheme, there are two possible conventions to which color is color 1. The more common conventions seem to be "black goes first", "red goes first", and "V goes first", but the opposite conventions also exist.

=== Board orientation ===

The board can be oriented in a number of different ways. It can be oriented in the ''positive'' or ''negative'' senses, and rotated to various angles.

'''Convention: board sense'''

Possible values:

* '''Positive:''' The cells A1, B1, A2 form a clockwise triangle in that order.

* '''Negative:''' The cells A1, B1, A2 form a counterclockwise triangle in that order.

The positive sense can also be described by saying that if the board is rotated so that A1 points west, then the ranks run from southwest to northeast, and the files run from northwest to southeast. This convention can also be described as "letters above numbers", because if A1 points left, the letters A,B,C marking the files appear above the numbers 1,2,3 marking the ranks.

'''Convention: board rotation'''

There is probably no need for a preferred board rotation; indeed, players using a physical game board may freely move around the board and look at it from any angle they want.

In computer hex, the most common board rotations are:

* '''Diamond:''' the A1 corner points west, i.e., in the direction of 9 o'clock on an analog clock.

* '''Flat:''' the A1 corner points northwest, i.e., in the direction of 10 o'clock.

* '''Flat II:''' the A1 corner points southwest, i.e., in the direction of 8 o'clock.

The following table illustrates some common board layouts. Here, color 1 is shown as black and color 2 is shown as white.

{| class="wikitable" style="text-align: center"
|-
| [[Image:Positive-Diamond.png|250px]]

Positive diamond
| [[Image:Positive-Flat.png|250px]]

Positive flat
| [[Image:Positive-Flat2.png|250px]]

Positive flat II
|-
| [[Image:Negative-Diamond.png|250px]]

Negative diamond
| [[Image:Negative-Flat.png|250px]]

Negative flat
| [[Image:Negative-Flat2.png|250px]]

Negative flat II
|}

== List of conventions used ==

{| class="wikitable"
|-
!
! Origin
! Edge coloring
! Swap
! Passing
! Colors
! First player
! Orientation
|-
| Browne
| acute
| normal
| sides
| N/A
| black/white
| black
| positive diamond
|-
| Seymour
| acute
| normal
| pieces
| explicit in puzzles, none in book
| black/white
| black
| positive diamond
|-
| Berge
| acute
| normal
| ?
| N/A
| black/white
| white
| negative flat II
|-
| Hayward and Toft
| acute
| ?
| sides
| N/A
| black/white
| any
| diamond
|-
| Hexy
| acute
| normal
| pieces
| none
| red/blue
| red
| positive diamond / positive flat / negative flat II
|-
| Mohex
| acute
| normal
| N/A
| implicit
| black/white
| black
| positive flat
|-
| HexGui
| acute
| normal
| both
| implicit
| black/white
| black
| any
|-
| Board Game Arena
| acute
| normal
| pieces
| explicit
| red/blue
| red
| positive flat
|-
| Little Golem
| acute
| normal
| pieces
| none
| black/white
| black
| positive flat
|-
| igGameCenter
| acute
| normal
| sides
| none
| red/blue or black/white
| red or black
| positive flat
|-
| PlayOK
| acute
| normal
| pieces
| none
| black/white
| black
| positive flat
|-
| PlayHex
| acute
| normal
| pieces
| explicit
| red/blue
| red
| any positive
|-
| AbstractPlay
| acute
| normal
| pieces
| none
| red/blue
| red
| positive flat
|-
| HexWorld
| acute
| normal
| both
| explicit
| black/white or red/blue
| black or red
| any positive
|-
| TRMPH
| acute
| normal
| N/A
| none
| black/white
| black
| positive flat
|-j
| HexWiki
| acute
| normal
| both
| N/A
| red/blue
| red
| positive flat
|}

Notes:

* Browne states that there is no universal convention for which color goes first, but black goes first in all examples in the book.

* Berge states that any player can go first, but white goes first in his example games.

* TRMPH implements swapping incorrectly. It changes the color of the piece, but not its location.

* MoHex does not implement a swap-pieces move. It implements swap-sides, which it incorrectly (and confusingly) calls swap-pieces.

* BoardGameArena has an explicit passing move, but three consecutive passes are not allowed. Therefore, if a player passes, the opponent can effectively reject the pass by passing too.

* AbstractPlay uses red and blue as the default colors, but users can set custom colors for themselves (these are not seen by the opponent). It uses the positive flat orientation and the board can be rotated in increments of 90 degrees.

[[category: Rules and Conventions]]

Conventions

2024-09-25T12:14:24Z

Alcalyn: orientatation

This page lists possible conventions for Hex games. These conventions concern the color of the players, who goes first, the orientation of the board, and the numbering of the positions on the board. Unlike much older games such as Chess and Go, there are no universally agreed upon conventions for Hex, and different people, game sites, books, and computer programs sometimes use different conventions.

We distinguish ''logical'' from ''physical'' conventions. Logical conventions are concerned with the abstract rules of Hex, irrespectively of how the players visualize the game. Physical conventions relate to the specific appearance of the game board. When the game is played remotely, for example on an internet game server or over email, it is in principle possible for the two players to follow different physical conventions; for example, each player may choose their own preferred orientation of the board, or their own preferred colors for the pieces. However, both players should follow the same logical conventions.

== Universal vs. local conventions ==

There are several aspects of a game of Hex that are arbitrary and do not affect game play in an essential way. This includes the orientation (rotation and reflection) of the board, the colors of the players, who goes first, which edge belongs to which player, and whether and how the cells on the board are numbered.

The same also applies to other games; for example, the game of Chess would not change in an essential way if the white square were in the bottom left corner instead of the bottom right one, if black went first instead of white, if the white queen started on a black square instead of a white one, if the ranks were lettered and the files numbered instead of the other way around, and so on.

In the case of established games such as Chess and Go, there are universal conventions that all players have agreed on. This has certain advantages. For example, every chess player understands what it means to move a white pawn from d2 to d4, without requiring further explanation.

In Hex, there are no universal conventions yet. Therefore, each player, book, game site, tournament, or hex program must define its own conventions. It may be desirable for a set of universal conventions to eventually crystallize.

== Logical conventions ==

=== Cell naming ===

Cells are arranged in a grid, with each cell named by a letter and a number. Letters can be uppercase or lowercase. The following convention seems to be nearly universally used:

[[Image:Logical-Board.png|right|]]

'''Convention: coordinate origin'''

Possible values:

* '''Acute:''' The cell A1 is an an acute corner of the board.

In particular, this means that the three cells A1, B1, and A2 are all adjacent to each other.

For the purpose of the following exposition, it is useful to refer to the cells that share a common number as a "rank", and to the cells that share a common letter as a "file", as in Chess. For example, the 1-rank consists of the cells A1, B1, C1, etc., and the A-file consists of the cells A1, A2, A3, etc.

Note that in some games, such as Go, it is customary to omit certain letters from the alphabet, especially the letter I. This is done "to avoid confusion between I and J", and presumably dates from a time when typesetting was uncommon and people had terrible handwriting. However, in Hex, the standard 26-letter alphabet is used. If more than 26 letters are needed, alphabet numbers can be used: the next file after Z is AA, then AB, and so on.

=== Edge coloring ===

To avoid referencing specific physical attributes of the game, we will refer to the color of the first piece played in the game as "color 1", and to the other color as "color 2". One pair of opposite edges "belongs to" color 1; these are the edges that the player who is playing color 1 is trying to connect. The other pair of opposite edges belongs to color 2.

'''Convention: edge coloring'''

Possible values:

* '''Normal:''' The edge that is adjacent to the 1-rank belongs to color 1, and the edge that is adjacent to the A-file belongs to color 2.

In other words, the normal edge coloring convention states that the color 1 edges are parallel to ranks, and the color 2 edges are parallel to files. In a typical rendering of the board, the letters (naming files) are written along the color 1 edge and the numbers (naming ranks) are written along the color 2 edge.

I am not aware of anybody who has used the opposite convention. However, there are some authors who use no edge coloring convention at all.

=== Swapping ===

There are two different ways of implementing the [[swap rule]]. Which convention is chosen will affect the notation for games. It is also possible to permit both methods of swapping; then it is up to the player to decide which method to use.

'''Convention: swap method'''

Possible values:

* '''Swap sides:''' Upon playing a swap move, the board position stays the same and the players change colors.

* '''Swap pieces:''' Upon playing a swap move, the players keep their colors and the board position is mirrored (ranks and files interchanged) and the color of the pieces is inverted. For example, a black piece at A2 would be replaced by a white piece at B1.

Here is an example using the swap sides convention: Player 1 plays a black piece at g4. Player 2 swaps sides. The board state remains unchanged, and immediately after the swap, it is player 1's turn to play a white piece.

Here is an example using the swap pieces convention: Player 1 plays a black piece at g4. Player 2 swaps pieces, and replaces the black piece at g4 with a white piece at d7. Immediately after the swap, it is player 1's turn to play a black piece.

When using algebraic notation for a sequence of moves, it is important to know which swap method was used. The swap method should either be defined for the context in which it is used, or else the notation should state explicitly which method was used. For example, with the swap sides convention, the notation "g4 swap f7 e7" means player 1 plays color 1 at g4, player 2 swaps, player 1 plays color 2 at f7, and player 2 plays color 1 at e7. With the swap pieces convention, the same game would be described as "g4 swap g6 g5". Here, player 1 plays color 1 at g4, player 2 replaces this by a piece by color 2 at d7, then player 1 plays color 1 at g6 and player 2 plays color 2 at g5. It is best not to use ambiguous notation at all; the two games can then be denoted "g4 swap-sides f7 e7" and "g4 swap-pieces g6 g5".

Note that the above logical conventions are purely symbolic. They do not depend on any particular board layout.

=== Passing ===

Although passing (skipping a move) is not always considered part of the classic Hex rules, allowing it does not change the nature of the game and has certain advantages. See the page on [[passing]] for more details.

'''Convention: passing'''

Possible values:

* '''Explicit''': Players can pass and there is an explicit passing move in the game record, e.g.: "Black passed".

* '''Implicit''': There is no explicit passing move, but a player can move twice in a row (presumably when the other player allows it).

* '''None''': Moves must be strictly alternating and passing is not allowed.

== Physical conventions ==

=== Colors ===

The most common color schemes are:

* Black and white.
* Red and blue.
* V and H.

With each color scheme, there are two possible conventions to which color is color 1. The more common conventions seem to be "black goes first", "red goes first", and "V goes first", but the opposite conventions also exist.

=== Board orientation ===

The board can be oriented in a number of different ways. It can be oriented in the ''positive'' or ''negative'' senses, and rotated to various angles.

'''Convention: board sense'''

Possible values:

* '''Positive:''' The cells A1, B1, A2 form a clockwise triangle in that order.

* '''Negative:''' The cells A1, B1, A2 form a counterclockwise triangle in that order.

The positive sense can also be described by saying that if the board is rotated so that A1 points west, then the ranks run from southwest to northeast, and the files run from northwest to southeast. This convention can also be described as "letters above numbers", because if A1 points left, the letters A,B,C marking the files appear above the numbers 1,2,3 marking the ranks.

'''Convention: board rotation'''

There is probably no need for a preferred board rotation; indeed, players using a physical game board may freely move around the board and look at it from any angle they want.

In computer hex, the most common board rotations are:

* '''Diamond:''' the A1 corner points west, i.e., in the direction of 9 o'clock on an analog clock.

* '''Flat:''' the A1 corner points northwest, i.e., in the direction of 10 o'clock.

* '''Flat II:''' the A1 corner points southwest, i.e., in the direction of 8 o'clock.

The following table illustrates some common board layouts. Here, color 1 is shown as black and color 2 is shown as white.

{| class="wikitable" style="text-align: center"
|-
| [[Image:Positive-Diamond.png|250px]]

Positive diamond
| [[Image:Positive-Flat.png|250px]]

Positive flat
| [[Image:Positive-Flat2.png|250px]]

Positive flat II
|-
| [[Image:Negative-Diamond.png|250px]]

Negative diamond
| [[Image:Negative-Flat.png|250px]]

Negative flat
| [[Image:Negative-Flat2.png|250px]]

Negative flat II
|}

== List of conventions used ==

{| class="wikitable"
|-
!
! Origin
! Edge coloring
! Swap
! Passing
! Colors
! First player
! Orientation
|-
| Browne
| acute
| normal
| sides
| N/A
| black/white
| black
| positive diamond
|-
| Seymour
| acute
| normal
| pieces
| explicit in puzzles, none in book
| black/white
| black
| positive diamond
|-
| Berge
| acute
| normal
| ?
| N/A
| black/white
| white
| negative flat II
|-
| Hayward and Toft
| acute
| ?
| sides
| N/A
| black/white
| any
| diamond
|-
| Hexy
| acute
| normal
| pieces
| none
| red/blue
| red
| positive diamond / positive flat / negative flat II
|-
| Mohex
| acute
| normal
| N/A
| implicit
| black/white
| black
| positive flat
|-
| HexGui
| acute
| normal
| both
| implicit
| black/white
| black
| any
|-
| Board Game Arena
| acute
| normal
| pieces
| explicit
| red/blue
| red
| positive flat
|-
| Little Golem
| acute
| normal
| pieces
| none
| black/white
| black
| positive flat
|-
| igGameCenter
| acute
| normal
| sides
| none
| red/blue or black/white
| red or black
| positive flat
|-
| PlayOK
| acute
| normal
| pieces
| none
| black/white
| black
| positive flat
|-
| HexWorld
| acute
| normal
| both
| explicit
| black/white or red/blue
| black or red
| any positive
|-
| TRMPH
| acute
| normal
| N/A
| none
| black/white
| black
| positive flat
|-j
| HexWiki
| acute
| normal
| both
| N/A
| red/blue
| red
| positive flat
|}

Notes:

* Browne states that there is no universal convention for which color goes first, but black goes first in all examples in the book.

* Berge states that any player can go first, but white goes first in his example games.

* TRMPH implements swapping incorrectly. It changes the color of the piece, but not its location.

* MoHex does not implement a swap-pieces move. It implements swap-sides, which it incorrectly (and confusingly) calls swap-pieces.

* BoardGameArena has an explicit passing move, but three consecutive passes are not allowed. Therefore, if a player passes, the opponent can effectively reject the pass by passing too.

[[category: Rules and Conventions]]

Hex theory

2024-06-06T21:23:51Z

Alcalyn: /* Winning strategy for non-square boards */ typo

Unlike many other games, some things can be said about '''[[Hex]]''' with absolute certainty. Whether this makes Hex a [[why did you start playing Hex|better game]] is of course debatable, but many find this attribute charming.

This page discusses some of the known properties of Hex.

== No draw ==

''Main article: [[Draw]].''

When a Hex board has been completely filled with stones, one and only one player has connected their edges. The proof idea is quite simple. On a full Hex board, consider the set ''A'' of all red cells that are connected to Red's top edge. If this set contains a cell on Red's bottom edge, then Red is the winner. Otherwise, Blue has a winning path by going along the boundary of ''A''.

Links to more detailed proofs are on [[Jack van Rijswijck|Javhar]]'s page [http://javhar1.googlepages.com/hexcannotendinadraw "Hex cannot end in a draw"].

== Winning Strategy ==

* When the [[Swap rule|swap option]] is not used, the [[Red (player)|first player]] has a [[winning strategy]].
* When playing with the swap option, the second player has a [[winning strategy]].
* On non-square boards, i.e., boards of size ''n''×''m'', where ''n''≠''m'', the player with the shorter distance to cover has a [[winning strategy]] regardless of who starts.

The first and third of these statements are proved below. The second statement is a simple consequence of the swap rule: since Hex has no draws, each move is either winning or losing. If the first player's opening move would be winning without the swap rule, the second player swaps and inherits the win. If the opening move would be losing, the second player declines to swap and goes on to win. Thus, the second player can always win.

=== Winning strategy for Red ===

In [http://en.wikipedia.org/wiki/Chess chess], while nobody seriously believes that Black (the second player) has a [[winning strategy]], nobody has been able to disprove it. On the other hand, in Hex, a simple strategy-stealing argument shows that the second player cannot have a [[winning strategy]], and therefore the first player must have one.

In fact, we can prove a more general statement: for boards of size ''n''×''n'', any position that is symmetric (i.e., invariant under reflection about the short or long diagonal and inverting the color of the pieces) is a winning position for the next player to move under optimal play. This follows from the fact that Hex is a monotone game: a position with additional pieces of a player's color is always at least as good for that player as the position without the additional pieces. If [[passing]] were allowed, it would therefore never be to a player's advantage to pass. If the second player to move had a winning strategy for a symmetric position, then the first player to move could simply steal that strategy by passing and therefore themselves becoming the second player to move. Since passing does not help the player, they also have a winning strategy without passing, contradicting the assumption that the other player was winning.

=== Winning strategy for non-square boards ===

Consider a board of size ''n'' × ''(n+1)''.
In this case, Blue has a shorter distance to cover than Red. Divide the board into two triangles as shown.
<hexboard size="6x5"
coords="none"
contents="S red:all blue:area(a6,e6,e2)
E 1:a1 2:b1 3:c1 4:d1 5:e1 6:a2 7:b2 8:c2 9:d2 10:a3 11:b3 12:c3 13:a4 14:b4 15:a5
E 1:e6 2:e5 3:e4 4:e3 5:e2 6:d6 7:d5 8:d4 9:d3 10:c6 11:c5 12:c4 13:b6 14:b5 15:a6"
/>
Consider the following [[pairing strategy]] for Blue: whenever Red plays in a numbered cell, Blue responds by playing in the other cell of the same number. After filling all of the cells in this way, Red cannot have a winning path. To see why, assume, for the sake of contradiction, that Red has a winning path. Consider a shortest such winning path from the top to the bottom edge, and consider the first point where the path crosses the boundary between the pink and blue triangles, for example as shown here:
<hexboard size="6x5"
coords="none"
contents="S red:all blue:area(a6,e6,e2)
E 1:a1 2:b1 3:c1 4:d1 5:e1 6:a2 7:b2 8:c2 9:d2 10:a3 11:b3 12:c3 13:a4 14:b4 15:a5
E 1:e6 2:e5 3:e4 4:e3 5:e2 6:d6 7:d5 8:d4 9:d3 10:c6 11:c5 12:c4 13:b6 14:b5 15:a6
R 3:c1 8:c2 11:b3 14:b4 12:c4"
/>
Note that the crossing must happen in the horizontal direction, since any two cells straddling the boundary in the diagonal direction have the same number and therefore cannot both be occupied by Red.
Since the red 14 is connected to the top by a red path within the pink triangle, by symmetry, the blue 14 is connected to the right by a blue path within the blue triangle:
<hexboard size="6x5"
coords="none"
contents="S red:all blue:area(a6,e6,e2)
E 1:a1 2:b1 3:c1 4:d1 5:e1 6:a2 7:b2 8:c2 9:d2 10:a3 11:b3 12:c3 13:a4 14:b4 15:a5
E 1:e6 2:e5 3:e4 4:e3 5:e2 6:d6 7:d5 8:d4 9:d3 10:c6 11:c5 12:c4 13:b6 14:b5 15:a6
R 3:c1 8:c2 11:b3 14:b4 12:c4
B 3:e4 8:d4 11:c5 14:b5"
/>
This cuts off 12 from the bottom. Indeed, the potential connection from 12 to the bottom cannot cross the path of blue stones from 14 to the right. It also cannot cross the path of red stones from 14 to the top (because then the red path would cross itself, contradicting our assumption that it was shortest). Therefore, the red path is "trapped" in the upper right area, showing that there can be no winning path for Red. It follows that Blue's pairing strategy is a winning strategy.

Since Blue has a second-player winning strategy, it follows that Blue also has a first-player winning strategy, because an additional move at the beginning cannot hurt Blue.

An analogous strategy works for all boards of size ''n''×''m'' with ''n'' > ''m''. If the difference between ''n'' and ''m'' is greater than 1, Blue can simply ignore the additional rows, say at the bottom of the board, i.e., pretend that they have already been filled with red stones. If Red moves in the ignored area, Blue can [[passing|pass]] (or in case passing is not permitted, Blue can move anywhere).

For symmetric reasons, Red has a winning strategy when ''n'' < ''m''.

See [[parallelogram boards]] for an analysis of how much head start the player with the larger distance needs to win, for different non-square boards.

== Complexity ==

* The problem of determining the winner of a given Hex position (on a board of size ''n''×''n'') is '''PSPACE-complete'''. The fact that it is a member of PSPACE is not surprising, because one can decide the winner by simply exploring the entire game tree, i.e., by playing every possible sequence of moves, which takes only a polynomial amount of memory. The fact that this decision problem is PSPACE-hard was first proved by [http://academic.timwylie.com/17CSCI4341/hex_acta.pdf Stefan Reisch] in 1979.

Several other related decision problems are also PSPACE-complete. For example:

* The detection of [[virtual connection]]s is PSPACE-complete. Clearly, this problem is in PSPACE, since one can decide the validity of a virtual connection by exploring every possible sequence of moves within the given carrier set. The fact that it is PSPACE-hard follows from the PSPACE-hardness of detecting a winning position, since a board position is winning (for the second player) if and only if it gives a virtual connection between the player's two edges.

* The detection of [[dominated cell]]s is PSPACE-complete. More specifically, given a board size, a position on that board, a player to move, and two empty cells X and Y, the problem of deciding whether X dominates Y is PSPACE-complete. To see why it is in PSPACE, note that it is sufficient to check for each of X and Y whether it is a winning move for the player in question. X dominates Y if and only if X is a winning move or Y is not a winning move. For PSPACE-hardness, consider the following position on a board of size (''n''+2)×(''n''+2), where the cells marked "*" denote some arbitrary position of an ''n''×''n'' board:<hexboard size="6x6" contents="R c2 d2 e2 f2 a2 a3 a4 a5 a6 B a1 d1 e1 f1 b2 b3 b4 b5 b6 E *:c3 *:c4 *:c5 *:c6 *:d3 *:d4 *:d5 *:d6 *:e3 *:e4 *:e5 *:e6 *:f3 *:f4 *:f5 *:f6"/>For Red, the move at b1 is clearly winning, whereas the move at c1 is winning if and only if Red has a first-player win in the game marked "*". Therefore, b1 dominates c1. Moreover, c1 dominates b1 if and only if Red has a first-player winning strategy for "*". Moreover, b1 ''strictly'' dominates c1 if and only if Red has no such strategy. Since answering the latter question on the ''n''×''n'' board is PSPACE-hard, it follows that both domination and strict domination are PSPACE-hard problems.

There are some computational problems in Hex that are easier than PSPACE.

* The problem of deciding whether a given cell is [[dead cell|dead]] is in co-NP. Equivalently, the problem of deciding whether a given cell is [[dead cell|alive]] is in NP. This is because an empty cell is alive if and only if it belongs to some minimal winning path, relative to the given board position. Given such a path, it is checkable in polynomial time whether it is winning and minimal. [https://webdocs.cs.ualberta.ca/~hayward/papers/bergeParis.pdf Björnsson et al.] showed that recognizing alive nodes is NP-complete in the [[Shannon game]], a generalization of Hex. It is not known whether recognizing alive cells in Hex is also NP-complete or whether it is easier.

== Solving Hex ==

* Hex has been solved on [[small boards]].

== See also ==

[[Open problems]]

== External links ==

* Stefan Reisch. [http://academic.timwylie.com/17CSCI4341/hex_acta.pdf Hex is PSPACE-complete], 1979.
* Stefan Kiefer. [https://web.archive.org/web/20070625134953/http://www.fmi.uni-stuttgart.de/szs/publications/info/kiefersn.Kie03.shtml Die Menge der virtuellen Verbindungen im Spiel Hex ist PSPACE-vollständig]. Studienarbeit Nr. 1887, Universität Stuttgart, Juli 2003. In German.
* Thomas Maarup. [https://web.archive.org/web/20221014000841/https://maarup.net/thomas/hex/ Hex]. Master's thesis, 2005.
* Yngvi Björnsson, Ryan Hayward, Michael Johanson, Jack Van Rijswijck. [https://webdocs.cs.ualberta.ca/~hayward/papers/bergeParis.pdf Dead cell analysis in Hex and the Shannon game]. Trends in Mathematics, pp.45–59, 2006.

[[category:Theory]]

PlayHex

2024-05-20T18:17:05Z

Alcalyn: put official max board limit, mason registered the name at same time but by coincidence

PlayHex is a website for playing Hex online. It supports all board sizes from 1x1 to 42x42, and flexible time controls for both real-time and turn-based play. It also provides AI opponents of various strengths, including a very strong version of [[KataHex]]. Moreover, the website offers almost instantaneous AI-based game analysis.

== Open source development ==

The PlayHex project, including the Hex server and client, is free software under an open source license. Its creator and principal developer is [[User:Alcalyn|Alcalyn]], with additional contributions by [[User:Comonoid|Comonoid]] and others. The source code is hosted on [https://github.com/alcalyn/hex/ GitHub].

== History ==

Development started in December 2022 and the playhex.org domain was registered at the same time, but independently, by [[User:Mason|Mason]]. On December 18, 2023, an early version of the still unnamed PlayHex website was made available to the community on the [[Hex_forums|Hex Discord Forum]]. The site went live at playhex.org in February 2024. As of May 2024, the project continues to be under active development.

== External links ==

* [https://playhex.org playhex.org]
* [https://github.com/alcalyn/hex/ GitHub repository for PlayHex]

[[category: Online play]]
[[category: Hex community]]

User:Alcalyn

2024-04-09T20:07:01Z

Alcalyn:

Hello!