Editing Hexapawn

{{Short description|Two-player game invented by Martin Gardner}}
{{#invoke:chessboard mxn|board|cols=3|rows=3
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| 3x3 Hexapawn setup
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{{Chess diagram 4x4
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| 4x4 Octopawn setup
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'''Hexapawn''' is a [[deterministic game|deterministic]] two-player [[game]] invented by [[Martin Gardner]]. It is played on a rectangular board of variable size, for example on a 3×3 board or on a regular [[chessboard]]. On a board of size ''n''×''m'', each player begins with ''m'' [[pawn (chess)|pawn]]s, one for each [[Square (geometry)|square]] in the row closest to them.  The goal of each player is to either advance a pawn to the opposite end of the board or leave the other player with no legal moves, either by [[stalemate]] or by having all of their pieces captured.

Hexapawn on the 3×3 board is a [[solved game]]; with perfect play, White will always lose in 3 moves (1.b2 axb2 2.cxb2 c2 3.a2 c1#). Indeed, Gardner specifically constructed it as a game with a small [[game tree]] in order to demonstrate how it could be played by a [[heuristic]] [[Artificial intelligence|AI]] implemented by a [[mechanical calculating device|mechanical computer]] based on [[Donald Michie]]'s [[Matchbox Educable Noughts and Crosses Engine]] (MENACE). 

A variant of this game is '''octopawn''', which is played on a 4×4 board with 4 pawns on each side. It is a forced win for White. 

Only 24 matchboxes are required for a hexapawn version of [[Matchbox Educable Noughts and Crosses Engine]].

==Rules==
As in [[chess]], a pawn may be moved in two different ways: it may be moved one square vertically forward, or it may capture a pawn one square diagonally ahead of it.  A pawn may not be moved forward if there is a pawn in the next square.  Unlike chess, the first move of a pawn may not advance it by two spaces.  A player loses if they have no legal moves or one of the other player's pawns reaches the end of the board.

==Dawson's chess==
Whenever a player advances a pawn to the penultimate rank and attacks an opposing pawn, there is a threat to proceed to the final rank by capture.  The opponent's only sensible responses, therefore, are to either capture the advanced pawn or advance the threatened one, the latter only being sensible in the case that there is one threatened pawn rather than two.  If one restricts 3×{{mvar|N}} hexapawn with the additional rule that capturing is always compulsory, the result is the game '''Dawson's chess'''. The game was invented by [[Thomas Rayner Dawson]] in 1935.<ref>{{Cite web |last=Ferguson |first=Thomas S. |title=A Note on Dawson's Chess |url=https://www.math.ucla.edu/~tom/papers/unpublished/DawsonChess.pdf }}</ref>

Dawson's chess reduces to the [[impartial game]] denoted '''.137''' in [[octal game#Game specification|Conway's notation]].  This means that it is equivalent to a [[Nim]]-like game in which:
*on a turn, the player may remove one to three objects from a heap,
*removing just one object is a legal move only if the removed object is the only object in the heap, and
*when removing three objects from a heap of five or more, the player may also split the remainder into two heaps.
The initial position is a single heap of size {{mvar|N}}.
The [[nim-sequence]] for this game is

 0.1120311033224'''0'''5'''22'''3301130211045'''2'''74'''0'''
   1120311033224455'''2'''33011302110453748
   1120311033224455933011302110453748
   1120311033224455933011302110453748
   1120311033224455933011302110453748 ...,

where bold entries indicate the values that differ from the eventual periodic behavior of the sequence.

==References==
{{reflist}}

==Sources==
* Mathematical Games, ''[[Scientific American]]'', March 1962, reprinted in ''The Unexpected Hanging and Other Mathematical Diversions'', by Martin Gardner, pp. 93ff

==External links==
* [https://web.archive.org/web/20050330222720/http://www.chessvariants.org/small.dir/hexapawn.html Hexapawn] - an article by Robert Price.
* [https://web.archive.org/web/20080616173331/http://www.javazoid.com/hexapawn.html Hexapawn java applet] - source code included.
* [https://web.archive.org/web/20160307154736/https://itunes.apple.com/us/app/hexapawn-game/id1084181644?mt=8 Hexapawn game for IOS]
* [https://behnamazizi.github.io/hexapawn/ Play Hexapawn] - Play Online

[[Category:Mathematical games]]
[[Category:Chess variants]]
[[Category:1962 in chess]]
[[Category:Board games introduced in 1962]]
[[Category:Solved games]]