222 



Mr. H. M. Taylor on the Relative Values 



when placed in succession on every square of the board, it 

 seems fair to assume that this gives a not very inexact measure 

 of the value of the piece. 



For special reasons the above problem is stated in the fol- 

 lowing manner : — " A king and a piece of different colours are 

 placed at random on two squares of a chessboard of n 2 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 different forms for odd and 

 even values of ft, the results are given merely for even values 

 of n, and the results for the ordinary chessboard of 64 squares 

 deduced from them. 



As the relative values of the knight and bishop on the ordi- 

 nary chessboard on this hypothesis came out in a ratio very dif- 

 ferent 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 would be in check but 

 unable to take the piece. This check is called safe check in 

 contradistinction to a mere check, which may be safe or unsafe 

 and which is called simple check. 



Simple check from one rook. 



A rook in any position checks 2(ft— 1) squares. The king 

 can be placed on ft 2 — 1 for any given position of the rook. 

 The chance of check, therefore, is 



2(w-l)_ 2 



If ft =8, 



ft 2 -l 



ft + 1 



the chance = § . 

 Safe check from one rook. 



a 



b 



b 





b 



c 



c 





b 



c 



c 













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 fol- 

 lollowing scheme : — 



