CH. VI] 



CHESS-BOARD RECREATIONS 



113 



check the other. The chance is 43/48, and therefore the chance 

 that they will occupy adjoining cells is 5/48. If three kings 

 are put on the board, the chance that no two of them occupy 

 adjoining cells is 1061/1488. The corresponding chances* for 

 a board of n a cells are (n — 1) (n - 2) (n a + 3n - 2)/n 2 (n? — 1) and 

 (w - 1) (w - 2) (m 4 + 3w 8 - 20re 2 - 30re + 132)/n s (n a - 1) (n 1 - 2). 



The Eight Queens Problem!. One of the classical 

 problems connected with a chess-board is the determination 

 of the number of ways in which eight queens can be placed on 

 a chess-board — or more generally, in which n queens can be 

 placed on a board of n 1 cells — so that no queen can take any 

 other. This was proposed originally by Franz Nauck in 1850. 



In 1874 Dr S. GuntherJ suggested a method of solution by 

 means of determinants. For, if each symbol represents the cor- 

 responding cell of the board^the possible solutions for a board 

 of n a cells are given by those terms, if any, of the determinant 



d t 



a? 



Chn— s "an— 3 

 Pm-a &2n— 1 



in which no letter and no suffix appears more than once. 



The reason is obvious. Every term in a determinant con- 

 tains one and only one element out of every row and out of 

 every column : hence any term will indicate a position on the 

 board in which the queens cannot take one another by moves 

 rook-wise. Again in the above determinant the letters and 

 suffixes are so arranged that all the same letters and all the 



* L'lntermidiaire dei MatMmaticiem, Paris, 1897, vol. iv, p. 6, and 1901, 

 vol. mi, p. 140. 



t On the history of this problem see W. Ahrens, Mathematische Unter- 

 haltungen und Spiele, Leipzig, 1901, chap. ix. 



X Grunert's Archiv der Mathematik und Physik, 1874, vol. lvi, pp. 281—292. 



B. ft 8 



