of the Pieces in Chess. 22 d 



n 1 



Put m = — -^ — j then the numerator of the chance-fraction 



— ■ 



= 4{^(n 2 -2«) + l-( w - 4 X n ~ 2 >-|}+2(»-l) 



= | l (»-l)(2»-l). 

 The chance ; therefore, for both odd and even values of n 

 2?(n-l)(2n-l) _ 2 gf| , 1 



If 71=8, 



the chance = ^. 



If the bishop be restricted to squares of one and the same colour/ 

 white or black, then if n be even the numerator and the deno- 

 minator of the chance-fraction are both halved, and the chance 

 is the same as before. If n be odd, we have two cases to con- 

 sider. If we take the squares of the same colour as the centre 



n—1 

 square, the numerator of the chance-fraction is twice „ 



terms of the series 



(n-l)(n-l) + (w + l)(n-3) + (n + 3)(n-5) + &c. 

 increased by 2(n — 1), 



If we take the squares of the other colour, the numerator of 



the chance-fraction is twice the — ~ — terms of the above series 



_ Q-l)(n + l)(2n-3) 

 3 

 Now, when a bishop occupies one square, the king can be 

 placed on any one of n 2 — 1 squares. Therefore the chance of 

 check when the bishop is restricted to squares of the same 

 colour as the middle one 



i (ri2 _ 1} B"(n+lX^+l)" 



2 



When the bishop is restricted to squares of the other colour, 

 the chance of check 



(w-l)(n+l)(2wT-3) 



3 J 2n-3 



n 2 — 1/9 i\ 3 n 2 — 1 



~2~ ,( ) 



Phil. Mag. S. 5.. Vol. 1. No. 3. March 1876. K 



