# Infinite chess

Last updated

Infinite chess is any variation of the game of 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

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, suggesting a setup with the chess pieces in the same relative positions as in classical chess, with knights replaced by nightriders and a rule preventing pieces from travelling too far from opposing pieces. [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]

## Decidability of short mates

For infinite chess, it has been found that the mate-in-n problem is decidable; that is, given a natural number n and a player to move and the positions (such as on ${\displaystyle \mathbb {Z} \times \mathbb {Z} }$) of a finite number of chess pieces that are uniformly mobile and with constant and linear freedom, there is an algorithm that will answer if there is a forced checkmate in at most n moves. [10] One such algorithm consists of expressing the instance as a sentence in Presburger arithmetic and using the decision procedure for Presburger arithmetic.

However, the winning-position problem is not known to be decidable. [10] In addition to the lack of an obvious upper bound on the smallest such n when there is a mate-in-n, there could also be positions for which there is a forced mate but no integer n such that there is a mate-in-n. For example, there could be a position such that after one move by black, the number of moves until black gets checkmated will equal the distance by which black moved whichever piece black moved.

## Variations

• 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. [11]
• Trappist-1: This variation uses the huygens, a chess piece that jumps prime numbers of squares, possibly preventing the game from ever being solved. [12] . This game feature excludes Trappist-1 from the proof that the mate-in-n problem is decidable.

## Related Research Articles

Shogi, also known as Japanese chess or the Game of Generals, is a two-player strategy board game native to Japan. In the same family as chess, makruk, shatranj, janggi and xiangqi, it is the most popular chess variant in Japan. Shōgi means general's board game.

The knight (♘,♞) is a piece in the game of chess, representing a knight. It is normally represented by a horse's head and neck. Each player starts with two knights, which begin on the row closest to the player, between the rooks and bishops.

Minishogi is a modern variant of shogi. The game was invented around 1970 by Shigenobu Kusumoto of Osaka, Japan. The rules are nearly identical to those of standard shogi, with the exception that it is played on a 5x5 board with a reduced number of pieces, and each player's promotion zone consists only of the rank farthest from the player.

A fairy chess piece, variant chess piece, unorthodox chess piece, or heterodox chess piece is a chess piece not used in conventional chess but incorporated into certain chess variants and some chess problems. Fairy pieces vary in the way they move. Because of the distributed and uncoordinated nature of unorthodox chess development, the same piece can have different names, and different pieces the same name in various contexts. Almost all are usually symbolised as inverted or rotated icons of the standard pieces in diagrams, and the meanings of these "wildcards" must be defined in each context separately. Pieces invented for use in chess variants rather than problems sometimes instead have special icons designed for them, but with some exceptions, many of these are not used beyond the individual games they were invented for.

Crazyhouse is a chess variant similar to bughouse chess, but with only two players. It effectively incorporates a rule from the game shogi, in which a player can introduce a captured piece back to the chessboard as their own.

Microshogi is a modern variant of shogi, with very different rules for promotion, and demotion. Kerry Handscomb of NOST gave it this English name. Although not confirmed, he credits its invention to the late Oyama Yasuharu, a top level shogi player. The game was invented before 1982.

Tsume shogi or tsume (詰め) is the Japanese term for a shogi miniature problem in which the goal is to checkmate the opponent's king. Tsume problems present a situation that might occur in a shogi game, and the solver must find out how to achieve checkmate. It is similar to a chess problem.

Judkins shogi is a modern variant of shogi, however it is not Japanese. Credit for its invention has been given to Paul Judkins of Norwich, UK, prior to April 1998.

Kyoto shogi is a modern variant of shogi. It was invented by Tamiya Katsuya c. 1976.

Heian dai shogi is an early large board variant of shogi as it was played in the Heian period. The same 12th century document which describes the Heian form of shogi also describes this variant. Unfortunately, this description does not give enough information to actually play the game, but this has not stopped people from attempting to reconstruct this early form of shogi. A fairly complete and playable reconstruction is outlined here.

Yonin shōgi,, is a four-person variant of shogi. It may be played with a dedicated yonin shogi set or with two sets of standard shogi pieces, and is played on a standard sized shogi board.

Sannin shōgi, or in full kokusai sannin shōgi, is a three-person shogi variant invented circa 1930 by Tanigasaki Jisuke and recently revived. It is played on a hexagonal grid of border length 7 with 127 cells. Standard shogi pieces may be used, and the rules for capture, promotion, drops, etc. are mostly similar to standard shogi. While piece movement differs somewhat from standard shogi, especially in the case of the powerful promoted king, the main difference in play is due to the rules for voluntary and mandatory alliance between two of the three players.

Three-player chess is a family of chess variants specially designed for three players. Many variations of three-player chess have been devised. They usually use a non-standard board, for example, a hexagonal or three-sided board that connects the center cells in a special way. The three armies are differentiated usually by color.

Shogi, like western chess, can be divided into the opening, middle game and endgame, each requiring a different strategy. The opening consists of arranging one's defenses and positioning for attack, the middle game consists of attempting to break through the opposing defenses while maintaining one's own, and the endgame starts when one side's defenses have been compromised.

Shatar and hiashatar are two chess variants played in Mongolia.

Solving chess means finding an optimal strategy for playing chess, i.e. one by which one of the players can always force a victory, or both can force a draw. It also means more generally solving chess-like games, such as infinite chess. According to Zermelo's theorem, a hypothetically determinable optimal strategy does exist for chess and chess-like games.

The following outline is provided as an overview of and topical guide to chess:

Joel David Hamkins is an American mathematician and philosopher based at the University of Oxford. He has made contributions in mathematical and philosophical logic, particularly set theory and the philosophy of set theory, in computability theory, and in group theory.

A chess variant is a game "related to, derived from, or inspired by chess". Such variants can differ from chess in many different ways.

## 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. 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 , available at arXiv : 1201.5597.
11. Chess on an infinite plane game rules.
12. Trappist-1 game rules