CH. Vl] CHESS-BOARD RECREATIONS 111 



A rook put on any cell commands 14 other cells. Wherever 

 the rook is placed there will remain 63 cells on which the king 

 may be placed, and on which it is equally likely that it 

 will be placed. Hence the chance of a simple check is 14/63, 

 that is, 2/9. Similarly on a board of n" cells the chance is 

 2 (« - l)/(n 2 - 1), that is, 2/(» + 1). 



A knight when placed on any of the 4 corner cells like 



11 commands 2 cells. When placed on any of the 8 cells like 



12 and 21 it commands 3 cells. When placed on any of the 

 4 cells like 22 or any of the 16 boundary cells like 13, 14, 15, 

 16, it commands 4 cells. When placed on any of the 16 cells 

 like 23, 24, 25, 26, it commands 6 cells. And when placed on 

 any of the remaining 16 middle cells it commands 8 cells. 

 Hence the average number of cells commanded by a knight put 

 on a chess-board is (4x2 + 8x3 + 20 x4 + 16x6 + 16 x 8)/64, 

 that is, 336/64. Accordingly if a king and a knight are put on 

 the board, the chance that the king will be in simple check 

 is 336/64 x 63, that is 1/12. Similarly on a board of m 2 cells 

 the chance is 8 (n — 2)/n 2 (n + 1). 



A bishop when placed on any of the ring of 28 boundary 

 cells commands 7 cells. When placed on any ring of the 

 20 cells next to the boundary cells, it commands 9 cells. 

 When placed on any of the 12 cells forming the next ring, 

 it commands 11 cells. When placed on the 4 middle cells 

 it commands 13 cells. Hence, if a king and a bishop are put 

 on the board the chance that the king will be in simple check 

 is (28 x 7 + 20 x 9 + 12 x 11 + 4 x 13)/64 x 63, that is, 5/36. 

 Similarly on a board of if cells, when n is even, the chance 

 is 2 (2n — l)/3w (n + 1). When n is odd the analysis is longer, 

 owing to the fact that in this case the number of white cells 

 on the board differs from the number of black cells. I do not 

 give the work, which presents no special difficulty. 



A queen when placed on any cell of a board commands all 

 the cells which a bishop and a rook when placed on that cell 

 would do. Hence, if a king and a queen are put on the board, 

 the chance that the king will be in simple check is 2/9 + 5/36, 

 that is, 13/36. Similarly on a board of n" cells, when n is even, 

 the chance is 2 (5w — l)/3n (ra + 1), 



