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Chess
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Chess
Chess

Chess is a two-player board game believed to have been played in India as early as the sixth century AD. In different parts of this world, different chess games are played. The most played variants are western chess, Shogi (in Japan), and Xiangqi (in China).

The western version of chess is a game played on an 8x8 board, called a chessboard, of alternating black and white squares. Pieces with different types of allowed moves are placed on the board, a set of black pieces in the first two rows and a set of white pieces in the last two rows. The pieces are called the bishop (2), king (1), knight (2), pawn (8), queen (1), and rook (2). The object of the game is to capture the opponent's king.

In Ingmar Bergman's 1958 film classic The Seventh Seal, a Knight and his squire arrive home from the crusades to find Black Death sweeping their country. As they approach home, Death appears to the knight and tells him it is his time. The knight then challenges Death to a chess game for his life. Chess also appears as one of the games known to the WOPR computer in the 1983 film WarGames.

Hardy (1999, p. 17) estimated the number of possible games of chess as 10^(10^(50)). In a game of 40 moves, the number of possible board positions P(40) is at least 10^(120) according to Peterson (1996). However, this value does not agree with the 10^(40) possible positions given by Beeler et al. (1972), which was obtained by estimating the number of pawn positions (in the no-captures situation, this is 15^8), multiplying by the possible positions for all pieces, then dividing by two for each of the (rook, knight) pairs that are interchangeable, and dividing by two for each pair of bishops (since half the positions will have the bishops on the same color squares). (However, note that there are more positions with one or two captures, since the pawns can then switch columns; Schroeppel 1996.) Shannon (1950) gave the estimate

P(40) approx (64!)/(32!(8!)^2(2!)^6) approx 10^(43).

Rex Stout's fictional detective Nero Wolfe quotes the number of possible games after ten moves as follows: "Wolfe grunted. One hundred and sixty-nine million, five hundred and eighteen thousand, eight hundred and twenty-nine followed by twenty-one ciphers. The number of ways the first ten moves, both sides, may be played" (Stout 1983). To be precise, the number of distinct chess positions after n moves for n==1, 2, ... are 20, 400, 5362, 71852, 809896?, 9132484?, ... (Schwarzkopf 1994, Sloane's A019319). The number of chess games that end in exactly n moves (including games that mate in fewer than n plies) for n==1, 2, 3, ... are 20, 400, 8902, 197742, 4897256, 120921506, 3284294545, ... (K. Thompson, Sloane's A006494).

Cunningham (1889) incorrectly found 197299 games and 71782 positions after the fourth move. C. Flye St. Marie was the first to find the correct number of positions after four moves: 71852. Dawson (1946) gives the source as Intermediare des Mathematiques (1895), but K. Fabel writes that Flye St. Marie corrected the number 71870 (that he found in 1895) to 71852 in 1903. The history of the determination of the chess sequences is discussed in Schwarzkopf (1994).

The analysis of chess is extremely complicated due to the many possible options at each move. Steinhaus (1999, pp. 11-14), as well as many entire books, consider clever end-game positions which may be analyzed completely.

Two problems in recreational mathematics ask the questions

1. How many pieces of a given type can be placed on a chessboard without any two attacking?

2. What is the smallest number of pieces needed to occupy or attack every square?

The answers are given for the usual 8x8 chessboard in the following table (Madachy 1979).

problemmax.min.
bishops problem148
kings problem169
knights problem3212
queens problem85
rooks problem88

SEE ALSO: Bishops Problem, Board, Checkers, Chessboard, Fairy Chess, Go, Gomory's Theorem, Hard Hexagon Entropy Constant, Kings Problem, Knights Problem, Knight's Tour, Magic Tour, Queens Problem, Rooks Problem, Shogi, Tour, Xiangqi. [Pages Linking Here]

REFERENCES:

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 124-127, 1987.

Beeler, M. et al. Item 95 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 35, Feb. 1972. http://www.inwap.com/pdp10/hbaker/hakmem/proposed.html#item95.

"The Chess Variant Pages." http://www.chessvariants.com/.

Culin, S. "Tjyang-keui--Chess." §82 in Games of the Orient: Korea, China, Japan. Rutland, VT: Charles E. Tuttle, pp. 82-91, 1965.

Dawson, T. R. "A Surprise Correction." The Fairy Chess Review 6, 44, 1946.

Dickins, A. "A Guide to Fairy Chess." p. 28, 1967/1969/1971.

Dudeney, H. E. "Chessboard Problems." Amusements in Mathematics. New York: Dover, pp. 84-109, 1970.

Elkies, N. D. "New Directions in Enumerative Chess Problems." Elec. J. Combin. 11, No. 2, A4, 2004. http://www.combinatorics.org/Volume_11/Abstracts/v11i2a4.html.

Ewerhart, C. "Backward Induction and the Game-Theoretic Analysis of Chess." Games and Economic Behavior 39, 206-214, 2002.

Fabel, K. "Nüsse." Die Schwalbe 84, 196, 1934.

Fabel, K. "Weihnachtsnüsse." Die Schwalbe 190, 97, 1947.

Fabel, K. "Weihnachtsnüsse." Die Schwalbe 195, 14, 1948.

Fabel, K. "Eröffnungen." Am Rande des Schachbretts, 34-35, 1947.

Fabel, K. "Die ersten Schritte." Rund um das Schachbrett, 107-109, 1955.

Fabel, K. "Eröffnungen." Schach und Zahl 8, 1966/1971.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.

Hunter, J. A. H. and Madachy, J. S. Mathematical Diversions. New York: Dover, pp. 86-89, 1975.

Kraitchik, M. "Chess and Checkers." §12.1.1 in Mathematical Recreations. New York: W. W. Norton, pp. 267-276, 1942.

Lasker, E. Lasker's Manual of Chess. New York: Dover, 1960.

Madachy, J. S. "Chessboard Placement Problems." Ch. 2 in Madachy's Mathematical Recreations. New York: Dover, pp. 34-54, 1979.

Parlett, D. S. Oxford History of Board Games. Oxford, England: Oxford University Press, 1999.

Pegg, E. Jr. "Math Games: Chessboard Tasks." Apr. 11, 2005. http://www.maa.org/editorial/mathgames/mathgames_04_11_05.html.

Peterson, I. "The Soul of a Chess Machine: Lessons Learned from a Contest Pitting Man against Computer." Sci. News 149, 200-201, Mar. 30, 1996.

Petkovic, M. Mathematics and Chess. New York: Dover, 1997.

Schroeppel, R. "Reprise: Number of Legal Chess Positions." tech-news@cs.arizona.edu posting, Aug. 18, 1996.

Schwarzkopf, B. "Die ersten Züge." Problemkiste, 142-143, No. 92, Apr. 1994.

Shannon, C. "Programming a Computer for Playing Chess." Phil. Mag. 41, 256-275, 1950.

Sloane, N. J. A. Sequences A006494, A007545/M5100, and A019319 in "The On-Line Encyclopedia of Integer Sequences."

Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 11-14, 1999.

Stout, R. "Gambit." In Seven Complete Nero Wolfe Novels. New York: Avenic Books, p. 475, 1983.

Velucchi, M. "Some On-Line PostScript MathChess Papers." http://anduin.eldar.org/~problemi/papers.html.

Watkins, J. Across the Board: The Mathematics of Chessboard Problems. Princeton, NJ: Princeton University Press, 2004.



LAST MODIFIED: March 23, 2006

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Weisstein, Eric W. "Chess." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Chess.html


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