110 CHESS-BOARD RECREATIONS [CH. VI 



n columns. Most of the problems which I shall describe can 

 be extended to meet the case of a board of n? cells. 



The usual chess-pieces are Kings, Queens, Bishops, Knights, 

 and Rooks or Castles; there are also Pawns. I assume that 

 the moves of these pieces are known to the reader. 



With the game itself and with chess problems of the usual 

 type I do not concern myself. Particular positions of the pieces 

 may be subject to mathematical analysis, but in general the 

 moves open to a player are so numerous as to make it im- 

 possible to see far ahead. Probably this is obvious, but it may 

 emphasize how impossible it is to discuss the theory of the 

 game effectively if I add that it has been shown that there 

 may be as many as 197299 ways of playing the first four moves, 

 and nearly 72000 different positions at the end of the first four 

 moves (two on each side), of which 16556 arise when the players 

 move pawns only*. 



Relative Value of Pieces. The first question to which 

 I will address myself is the determination of the relative values 

 of the different chess-pieces f. 



If a piece is placed on a cell, the number of cells it com- 

 mands depends in general on its position. We may estimate 

 the value of the piece by the average number of cells which it 

 commands when placed in succession on every cell of the board. 

 This is equivalent to saying that the value of a piece may 

 be estimated by the chance that if it and a king are put at 

 random on the board, the king will be in check : if no other re- 

 striction is imposed this is called a simple*check. On whatever 

 cell the piece is originally placed there will remain 63 other 

 cells on which the king may be placed. It is equally probable 

 that it may be put on any one of them. Hence the chance 

 that it will be in check is 1/63 of the average number of cells 

 commanded by the piece. 



* L'Intermediaire des Mathematiciens, Paris, December, 1903, vol. x, pp. 

 305 — 308 : also Royal Engineers Journal, London, August — November, 1889 ; 

 or British Association Transactions, 1890, p. 745. 



+ H. M. Taylor, Philosophical Magazine, March, 1876, series 5, vol. I, 

 pp. 221—229. 



