Chess
Chess is a game played on an 
board, called a Chessboard, of alternating black and white squares. Pieceswith different types of allowed moves are placed on the board, a set of black pieces in the first two rows and a set of whitepieces 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. It is believed that chess was played in India as early as thesixth century AD. 
In a game of 40 moves, the number of possible board positions is at least 10 The number of chess games which end in exactly Cunningham (1889) incorrectly found 197,299 games and 71,782 positions after the fourth move. C. Flye St. Marie was thefirst to find the correct number of positions after four moves: 71,852. Dawson (1946) gives the source as Intermediare des Mathematiques (1895), but K. Fabel writes that Flye St. Marie corrected the number 71,870 (which he foundin 1895) to 71,852 in 1903. The history of the determination of the chess sequences is discussed in Schwarzkopf (1994). Two problems in recreational mathematics ask Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 124-127, 1987. Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Feb. 1972. 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. 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. 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. Madachy, J. S. ``Chessboard Placement Problems.'' Ch. 2 in Madachy's Mathematical Recreations. New York: Dover, pp. 34-54, 1979. 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.A006494,A019319, andA007545/M5100in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html. Stout, R. ``Gambit.'' In Seven Complete Nero Wolfe Novels. New York: Avenic Books, p. 475, 1983.
), times allpieces in all positions, dividing by 2 for each of the (rook, knight) which are interchangeable, dividing by 2 for eachpair of bishops (since half the positions will have the bishops on the same color squares). There are more positions with oneor two captures, since the pawns can then switch columns (Schroeppel 1996). Shannon (1950) gave the value
plies (including games that mate in fewer than plies) for 
, 2,3, ... are 20, 400, 8902, 197742, 4897256, 120921506, 3284294545, ... (K. Thompson, Sloane's A006494). Rex Stout'sfictional detective Nero Wolfe quotes the number of possible games after ten moves as follows: ``Wolfe grunted. Onehundred and sixty-nine million, five hundred and eighteen thousand, eight hundred and twenty-nine followed by twenty-oneciphers. The number of ways the first ten moves, both sides, may be played'' (Stout 1983). The number of chess positionsafter moves for , 2, ... are 20, 400, 5362, 71852, 809896?, 9132484?, ... (Schwarzkopf 1994,Sloane's A019319).
The answers are given in the following table (Madachy 1979).Piece Max. Min. Bishops 14 8 Kings 16 9 Knights 32 12 Queens 8 5 Rooks 8 8
References
- Law of Cancellation
- Law of Cosines
- Law of Large Numbers
- Law of Sines
- Law of Small Numbers
- Law of Tangents
- Law of Truly Large Numbers
- Lax-Milgram Theorem
- Lax Pair
- Layer
- Leading Digit Phenomenon
- Leading Order Analysis
- Leaf (Foliation)
- Leaf (Tree)
- Leakage
- Least Bound
- Least Common Multiple
- Least Deficient Number
- Least Period
- Least Prime Factor
- Least Squares Fitting
- Least Squares Fitting--Exponential
- Least Squares Fitting--Logarithmic
- Least Squares Fitting--Power Law
- Least Upper Bound