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 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 . The number of possible games of 40 moves or less is approximately (Beeler et al. 1972), a number arrived at by estimating the number of pawn positions (in the no-captures situation, this is ), 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

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 moves for , 2, ... are 20, 400, 5362, 71852, 809896?, 9132484?, ... (Schwarzkopf 1994, OEIS A019319). The number of chess games that end in exactly moves (including games that mate in fewer than plies) for , 2, 3, ... are 20, 400, 8902, 197742, 4897256, 120921506, 3284294545, ... (OEIS A006494).

Cunningham (1889) incorrectly found games and positions after the fourth move. C. Flye St. Marie was the first to find the correct number of positions after four moves: . Dawson (1946) gives the source as Intermediare des Mathematiques (1895), but K. Fabel writes that Flye St. Marie corrected the number (that he found in 1895) to 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 chessboard in the following table (Madachy 1979).