120 CHESS-BOARD RECREATIONS [CH. VI 



For queens the problem has been also discussed for a board 

 of n s cells where n has various values*. One queen can be 

 placed so as to command all the cells when n = 2 or 3, and 

 there is only 1 fundamental solution. Two queens are required 

 when n = 4 ; and there are 3 fundamental solutions, namely, 

 when they are placed on the cells 11 and 33, or on the cells 12 

 and 42, or on the cells 22, 23: these give 12 solutions in all. 

 Three queens are required when n = 5 ; and there are 37 funda- 

 mental solutions, giving 186 solutions in all. Three queens 

 are also required when n = 6, but there is only 1 fundamental 

 solution, namely, when they are put on the cells 11, 35, and 

 53, giving 4 solutions in all. Four queens are required when 

 rc = 7, one solution is when they are put on the cells 12, 26, 

 41, 55. 



Jaenisch proposed also the problem of the determination of 

 the minimum number of queens which can be placed on a board 

 of n s cells so as to command all the unoccupied cells, subject to 

 the restriction that no queen shall attack the cell occupied by 

 any other queen. In this case three queens are required when 

 n = 4, for instance, they can be put on the cells 11, 23, 42 ; and 

 there are 2 fundamental solutions, giving 16 solutions in all. 

 Three queens are required when n = 5, for instance, they can be 

 put on the cells 11, 24, 43, or on the cells 11, 34, 53; and there 

 are 2 fundamental solutions in all. Four queens are required 

 when n = 6, for instance, when they are put on the cells 13, 36, 

 41, 64 ; and there are 17 fundamental solutions. Four queens 

 are required when n = 7, and there is only 1 fundamental solu- 

 tion, namely, that already mentioned, when they are put on the 

 cells 12, 26, 41, 55, which gives 8 solutions in all. Five queens 

 are required when n = 8, and there are no less than 91 funda- 

 mental solutions ; for instance, one is when they are put on the 

 cells 11, 23, 37, 62, 76. 



I leave to any of my readers who may be interested in such 

 questions the discussion of the corresponding problems for the 



* C. P. de Jaenisch, Applications de V Analyse MatMmatique au Jeu des 

 tichecs, Petrograd, 1862, Appendix, p. 244 et seq. ; see also L'lntermidiaire 

 det MatMmaticiens, Paris, 1901, vol. vm, p. 88. 



