of the Pieces in Chess. 



227 



Therefore the chance 



Ifn=8, 



f(tt-2)Q-l)(5n-3) _ 2 (w-2)(5n-3) 

 n\n 2 -l) 3 n 2 (n+l) ' 



the chance = 1 ^£. 

 Simple check with two bishops. 

 We will assume that n is an even number, and that one 

 bishop (A) is restricted to white squares, and the other (B) to 

 black squares. The number of squares checked by the two 

 bishops for all different positions of B when A is on a parti- 

 cular square is obtained from the following scheme : — 



III a ' III a ' li 



aJ II 



11 v I li 



11 "' 11 C ' 1 



a' 



ill c ' III 



llll] 





|lllll|l!ffl!ll| III 1 ' 1 " 



A on 



a square on the (^— r\th row from the outside. 



Numher of 

 such squares. 



2(2r-l) 



Bon 



a 

 b 

 c 

 d 



Number of 



sueh positions 



ofB. 



2(71-1) 



2(n-3) 

 2(rc-5) 

 2(n-7) 



Number of squares 

 checked by A 



and B. 

 Sn-2r-2 

 Sn — 2r 

 3n—2r + 2 

 Sn-2r + 4= 



The total number of checks for all positions of B while A 

 remains fixed 



= 2(n-l)(3n-2r-2) + 2(n-3)(3rc-2r) 



+ 2(n-5)(3n-2r + 2) + 2(n-7)(3n-2r + 4) + &c. to 



x terms, 

 fit 



= 2(ri-l)((2ri-2r-3) + n + l) 



+ 2(n-3)((2n-2r-3) + n + 3) 



+ 2(ri-5)((2n~-2^-3) + w + 5)+ &c. to ^ terms, 



= 2.f(2n-2^3) + 2g.^-!<^L).) 



= _ rn 2 + ^ 10n 2_ 9n+2 ). 



R2 



