112 CHESS-BOAKD RECREATIONS [CH. VI 



On the above assumptions the relative values of the rook, 

 knight, bishop, and queen are 16, 6, 10, 26. According to 

 Staunton's Chess-Player's Handbook the actual values, estimated 

 empirically, are in the ratio of 548, 305, 350, 994 ; according to 

 Von Bilguer the ratios are 540, 350, 360, 1000 — the value of a 

 pawn being taken as 100. 



There is considerable discrepancy between the above results 

 as given by theory and practice. It has been, however, sug- 

 gested that a better test of the value of a piece would be the 

 chance that when it and a king were put at random on the 

 board it would check the king without giving the king the 

 opportunity of taking it. This is called a safe check as distin- 

 guished from a simple check. 



Applying the same method as above, the chances of a safe 

 check work out as follows. For a rook the chance of a safe 

 check is (4 x 12 + 24 x 11 + 36 x 10)/64 x 63, that is, 1/6 ; or 

 on a board of n 2 cells is 2 (n — 2)/n (n + 1). For a knight all 

 checks are safe, and therefore the chance of a safe check is 1/12 ; 

 or on a board of v? cells is 8 (n — 2)/*i a (n + 1). For a bishop 

 the chance of a safe check is 364/64 x 63, that is, 13/144; or on a 

 board of w 2 cells, when n is even, is 2 (n — 2) (2n — 3)/3» 2 (n + 1). 

 For a queen the chance of a safe check is 1036/64 x 63, that is, 

 37/144 ; or on a board of w 2 cells, is 2 (n - 2) (5n - 3)/3w 2 (n + 1), 

 when n is even. 



On this view the relative values of the rook, knight, bishop, 

 queen are 24, 12, 13, 37 ; while, according to Staunton, experi- 

 ence shows that they are approximately 22, 12, 14, 40, and 

 according to Von Bilguer, 18, 12, 12, 33. 



The same method can be applied to compare the values 

 of combinations of pieces. For instance the value of two 

 bishops (one restricted to white cells and the other to black 

 cells) and two rooks, estimated by the chance of a simple 

 check, are respectively 35/124 and 37/93. Hence on this view 

 a queen in general should be more valuable than two bishops 

 but less valuable than two rooks. This agrees with experience. 



An analogous problem consists in finding the chance that 

 two kings, put at random on the board, will not occupy adjoin- 

 ing cells, that is, that neither would (were such a move possible) 



