Hexapawn is a

deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and consi ...

two-player

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

invented by

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

. 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 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 play, the bo ...

. 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 piece in the game * Pawnbroker or pawnshop, a business that provides loans by taking personal property as collateral Pawn may also refer to: Places * Pawn, Oregon, an his ...

s, one for each

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

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. Hexapawn on the 3×3 board is a

solved game A solved game is a game whose outcome (win, lose or 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 full informa ...

; 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, which typically studies sequential games with perfect information, a game tree is a graph representing all possible game states within such a game. Such games include well-known ones such as chess, check ...

in order to demonstrate how it could be played by a

heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...

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 increment outp ...

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 ...

Matchbox Educable Noughts and Crosses Engine The Matchbox Educable Noughts and Crosses Engine (sometimes called the Machine Educable Noughts and Crosses Engine or MENACE) was a mechanical computer made from 304 matchboxes designed and built by artificial intelligence researcher Donald Mic ...

. 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.

Rules

As in

chess Chess is a board game for two players, called White and Black, each controlling an army of chess pieces in their color, with the objective to checkmate the opponent's king. It is sometimes called international chess or Western chess to disti ...

, 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. 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 betw ...

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 {{mvar, N. 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

* Mathematical Games, ''

Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...

'', 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