CH. VI] CHESS-BOARD RECREATIONS 119 



and so on. The rule is obvious except when n is of the form 

 6m + 2 or 6m + 3. 



Maximum Pieces Problem*. The Eight Queens Problem 

 suggests the somewhat analogous question of finding the 

 maximum number of kings — or more generally of pieces of 

 one type — which can be put on a board so that no one can 

 take any other, and the number of solutions possible in each 

 case. 



In the case of kings the number is 16; for instance, one 

 solution is when they are put on the cells 11, 13, 15, 17, 31, 33, 

 35, 37, 51, 53, 55, 57, 71, 73, 75, 77. For queens, it is obvious 

 that the problem is covered by the analysis already given, and 

 the number is 8. For bishops the number is 14, the pieces 

 being put on the boundary cells; for instance one solution is 

 when they are put on the cells 11, 12, 13, 14, 15, 16, 17, 81, 82, 

 83, 84, 85, 86, 87, there are 256 solutions. For knights the 

 number is 32 ; for instance, they can be put on all the white 

 or on all the black cells, and there are 2 fundamental solutions. 

 For rooks it is obvious that the number is 8, and there are in 

 all 8 ! solutions. 



Minimum Pieces Problem*. Another problem of a some- 

 what similar character is the determination of the minimum 

 number of kings — or more generally of pieces of one type — 

 which can be put on a board so as to command or occupy all 



the cells. 



For kings the number is 9; for instance, they can be put on 

 the cells 11, 14, 17, 41, 44, 47, 71, 74, 77. For queens the 

 number is 5; for instance, they can be put on the cells 18, 35, 

 41, 76, 82. For bishops the number is 8; for instance, they can 

 be'put on the cells 41, 42, 43, 44, 45, 46, 47, 48. For knights 

 the number is 12; for instance, they can be put on the cells 

 26, 32, 33, 35, 36, 43, 56, 63, 64, 66, 67, and 73— constituting 

 four triplets arranged symmetrically. For rooks the number 

 is 8, and the solutions are obvious. 



* Mr H. E. Dudeney has written on these problems in the Weekly Dispatch. 



