22 



KEPORT 1875. 



ber of squares which any particular piece commands when placed in succession on 

 every square of the hoard, it seems fair to assume that this gives a not very inexact 

 measm-e of the value of the piece. 



For special reasons the above problem is stated in the following manner : — " A 

 kino- and a piece of different colours are placed at random on two squares of a chess- 

 hoard of tx^ squares : it is required to find the chance that the king is in check." 

 The ordinary chessboard has an even number of squares ; and as some of the results 

 take difierent forms for odd and even values of n, the residts are here given merely 

 for even values of w, and the results for the ordinary chessboard of sixty-four squares 

 deduced from them. As the relative values of the knight and bishop on the ordi- 

 nary chessboard on this hypothesis come out in a ratio very different from the ratio 

 that is ordinarily received by chess-players, it occurred to the author to investigate 

 the chance that 'when a king and a piece of different colours were placed at random 

 on two squares of a board, the king should be in check but unable to take the 

 piece. This check is called safe check in distinction to a mere check, which may 

 be safe or unsafe, which is called simple check. 



Simple check from one rook. — A rook on any position checks 2(m — 1) squares. 

 The king can be" placed on w^ — 1 squares for any given position of the rook. The 



chance of check, therefore, is ^^, — —^ = — -^ . If «=8, the chance = =. 

 ' ' n^ — 1 m4-1 9 



Safe check from one rook. — If the rook be on a corner square, it could be taken 



by a king in check on two squares, and so on. The number of safe checks by a 



rook on the different squares is given by the following scheme : — 



Rook on 

 a 



b 



c 



Number of 



safe cbecks. 



2w-4 



2n-5 



2n-Q 



Number of such 

 positions of the rook. 



4 

 4(h-2) 



(n-2f 



The chance 



4(-2n-i)+4(n-2)(2n-o) + (n-2f(2n-6) _ 2(w-2) 

 «-(«^— 1) 

 1 



«(w + l)' 



When « = 8, the chance = ,. 



Sitnple check with one hiight. — The number of squares attacked by a knight placed 

 on any square of a chessboard is given by the following scheme : — 



The chance of check 



^ 2 . 4-h3 ■ 8+4 . 4(m- 3) + 6 . A{n-4)+%(n ~Af _ 8( »-2) 



1 



If M=8, chance = 



12' 



For a knight all checks are safe checka. 



