# Infinite chess

Jianying Ji's infinite chess scheme, represented by ASCII characters (2000).

Infinite chess is any of several variations of the game chess played on an unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathematicians, both as a playable game and as a model for theoretical study, it has been found that even though the board is unbounded, there are ways in which a player can win the game in a finite number of moves.

## Background

Taikyoku shōgi (36×36 squares)

Classical (FIDE) chess is played on an 8×8 board (64 squares). However, the history of chess includes variants of the game played on boards of various sizes. A predecessor game called Courier chess was played on a slightly larger 12×8 board (96 squares) in the 12th century, and continued to be played for at least six hundred years. Japanese chess (shogi) has been played historically on boards of various sizes; the largest is taikyoku shōgi ("ultimate chess"). This chess-like game, which dates to the mid 16th century, was played on a 36×36 board (1296 squares), each player starts with 402 pieces of 209 different types, and a well-played game would require several days of play, possibly requiring each player to make over a thousand moves.[1][2][3][4]

Chess player Jianying Ji was one of many to propose infinite chess; in 2000, he suggested a setup with the chess pieces in the same relative positions as in classical chess.[5] Numerous other chess players, chess theorists, and mathematicians who study game theory have conceived of variations of infinite chess, often with different objectives in mind. Chess players sometimes use the scheme simply to alter the strategy; since chess pieces, and in particular the king, cannot be trapped in corners on an infinite board, new patterns are required to form a checkmate. Theorists conceive of infinite chess variations to expand the theory of chess in general, or as a model to study other mathematical, economic, or game-playing strategies.[6][7][8][9]

## In context of game theory

For infinite chess, mathematical investigations have shown that in a general endgame, one player can force a win in a finite number of moves. More specifically, it has been found that infinite chess is decidable; that is, given a position (such as ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$, and assuming pawns do not promote) of a finite number of chess pieces which are uniformly mobile and with constant and linear freedom, and (for example) White to move, there is an algorithm that will answer if White can win or force a draw, against any defense by Black.[10][11]

## Variations

Chess on an infinite plane starting position: guards are on (1,1),(8,1),(1,8),(8,8); hawks are on (−2,−6),(11,−6),(−2,15),(11,15); chancellors are on (0,1), (9,1), (0,8), (9,8)
• Chess on an infinite plane: 76 pieces are played on an unbounded chessboard. The game uses orthodox chess pieces, plus guards, hawks, and chancellors, the absence of borders makes pieces effectively less powerful (as the king and other pieces cannot be trapped in corners), so the added material helps compensate for this.[12]
• Trappist-1: This variation uses the huygens, a chess piece that jumps prime numbers of squares, possibly preventing the game from ever being solved.[13][nb 1]

## Notes

1. ^ This game feature excludes Trappist-1 from the mathematical conclusion of decidability.[10][11]

## References

1. ^ boardgamegeek/taikyoku-shogi boardgamegeek/taikyoku-shogi.
2. ^ chessvariants.com/taikyoku-shogi chessvariants.com/taikyoku-shogi.
3. ^ abstractstrategygames/ultimate-battle-chess.html abstractstrategygames/ultimate-battle-chess.
4. ^ history.chess.taishogi history.chess/taishogi.
5. ^ Infinite Chess at The Chess Variant Pages. An infinite chess scheme represented using ASCII characters.
6. ^ "Infinite Chess, PBS Infinite Series" PBS Infinite Series, with academic sources by J. Hamkins (infinite chess: https://arxiv.org/abs/1302.4377 and https://arxiv.org/abs/1510.08155).
7. ^ Aviezri Fraenkel; D. Lichtenstein (1981), "Computing a perfect strategy for n×n chess requires time exponential in n", J. Combin. Theory Ser. A, 31 (2): 199–214, doi:10.1016/0097-3165(81)90016-9
8. ^ Transfinite Game Values in Infinite Chess) Transfinite Game Values in Infinite Chess, 2014, (C.D.A. Evans, Joel Hamkins).
9. ^ "A position in infinite chess with game value w^4" Transfinite game values in infinite chess, January 2017; A position in infinite chess with game value w^4, October 2015; An introduction to the theory of infinite games, with examples from infinite chess, November 2014; The theory of infinite games: how to play infinite chess and win, August 2014; and other academic papers by Joel Hamkins.
10. ^ a b Mathoverflow.net/Decidability-of-chess-on-an-infinite-board Decidability-of-chess-on-an-infinite-board
11. ^ a b Dan Brumleve, Joel David Hamkins, Philipp Schlicht, The Mate-in-n Problem of Infinite Chess Is Decidable, Lecture Notes in Computer Science, Volume 7318, 2012, pp. 78-88, Springer [1], available at arXiv.
12. ^ Chess on an infinite plane game rules.
13. ^ Trappist-1 game rules