136 CHESS-BOARD RECREATIONS [CH. VI 



propounded in 1512. It was quoted by Lucas in 1894, but I 

 believe has not been published otherwise than in his works and 

 the earlier editions of this book. On a board of nine cells, such 

 as that drawn below, the two white knights are placed on the 



two top corner cells (a, d), and the two black knights on the 

 two bottom corner cells (6, c) : the other cells are left vacant. It 

 is required to move the knights so that the white knights shall 

 occupy the cells b and c, while the black shall occupy the cells 

 a and d. The solution is obvious. 



Queens' Problem. Another problem consists in placing 

 sixteen queens on a board so that no three are in a straight 

 line *. One solution is to place them on the cells 15, 16, 25, 26, 

 31, 32, 41, 42, 57, 58, 67, 68, 73, 74, 83, 84. It is of course 

 assumed that each queen is placed on the middle of its cell. 



Latin Squares. Another problem of the chess-board type 

 is the determination of the number x n of Latin Squares of 

 any assigned order n: a Latin Square of the nth order being 

 denned as a square of n 2 cells (in n rows and n columns) in 

 which n a letters consisting of n "a's," n "b's,"..., are arranged in 

 the cells so that the n letters in each row and each column are 

 different. The general theory is difficult f, but it may amuse 

 my readers to verify the following results for some of the lower 

 values of n : x 2 = 2, os 3 = 6, a? 4 = 576, x s = 149760. Clearly x n 

 is a multiple of n ! (n — 1) ! 



* H. E. Dudeney, The Tribune, November 7, 1906. 



+ See P. A. MaeMahon, Combinatory Analysis, Cambridge, 1915-16, vol. I, 

 pp. 246—263 ; vol. n, pp. 323—326. 



