Hexapawn is a

deterministic Determinism is the metaphysical view that all events within the universe (or multiverse) can occur only in one possible way. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping mo ...

two-player

game A game is a structured type of play usually undertaken for entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator sports or video games) or art ...

invented by

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writin ...

. It is played on a rectangular board of variable size, for example on a 3×3 board or on a regular

chessboard A chessboard is a game board used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During p ...

. On a board of size ''n''×''m'', each player begins with ''m''

pawn Pawn most often refers to: * Pawn (chess), the weakest and most numerous chess piece in the game * Pawnbroker or pawnshop, a business that provides loans by taking personal property as collateral Pawn or The Pawn may also refer to: Places * Pa ...

s, one for each

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

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 Stalemate is a situation in chess where the player whose turn it is to move is not in check and has no legal move. Stalemate results in a draw. During the endgame, stalemate is a resource that can enable the player with the inferior position ...

or by having all of their pieces captured. Hexapawn on the 3×3 board is a

solved game A solved game is a game whose outcome (win, lose or tie (draw), draw) can be correctly predicted from any position, assuming that both players play perfectly. This concept is usually applied to abstract strategy games, and especially to games with ...

; 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 the context of combinatorial game theory, a game tree is a graph representing all possible game states within a sequential game that has perfect information. Such games include chess, checkers, Go, and tic-tac-toe. A game tree can be us ...

in order to demonstrate how it could be played by a

heuristic A heuristic or heuristic technique (''problem solving'', '' mental shortcut'', ''rule of thumb'') is any approach to problem solving that employs a pragmatic method that is not fully optimized, perfected, or rationalized, but is nevertheless ...

AI implemented by a

mechanical computer A mechanical computer is a computer built from mechanical components such as levers and gears rather than electronic components. The most common examples are adding machines and mechanical counters, which use the turning of gears to incremen ...

based on

Donald Michie Donald Michie (; 11 November 1923 – 7 July 2007) was a British researcher in artificial intelligence. During World War II, Michie worked for the Government Code and Cypher School at Bletchley Park, contributing to the effort to solve " Tunny ...

'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 Chess is a board game for two players. It is an abstract strategy game that involves Perfect information, no hidden information and no elements of game of chance, chance. It is played on a square chessboard, board consisting of 64 squares arran ...

, 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× 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 Thomas Rayner Dawson (28 November 1889 – 16 December 1951) was an English chess problemist and is acknowledged as "the father of Fairy Chess". He invented many fairy pieces and new conditions. He introduced the popular fairy pieces grassho ...

in 1935. Dawson's chess reduces to the

impartial game In combinatorial game theory, an impartial game is a game in which the allowable moves depend only on the position and not on which of the two players is currently moving, and where the payoffs are symmetric. In other words, the only difference be ...

denoted .137 in 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 . The nim-sequence for this game is 0.1120311033224052233011302110452740 1120311033224455233011302110453748 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 ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Pri ...

'', March 1962, reprinted in ''The Unexpected Hanging and Other Mathematical Diversions'', by Martin Gardner, pp. 93ff

External links

Hexapawn
- an article by Robert Price.

- source code included.
Hexapawn game for IOS

Play Hexapawn
- Play Online Mathematical games Chess variants 1962 in chess Board games introduced in 1962 Solved games