CH. VI] CHESS-BOARD RECREATIONS 121 



other pieces*, and of the number of possible solutions in each 

 case. 



A problem of the same nature would be the determination 

 of the minimum number of queens (or other pieces) which 

 can be placed on a board so as to protect one another and 

 command all the unoccupied cells. For queens the number 

 is 5; for instance, they can be put on the cells 24, 34, 44, 

 54 and 84. For bishops the number is 10; for instance, they 

 can be put on the cells 24, 25, 34, 35, 44, 45, 64, 65, 74, and 75. 

 For knights the number is 14; for instance, they can be put on 

 the cells 32, 33, 36, 37, 43, 44, 45, 46, 63, 64, 65, 66, 73, and 76: 

 the solution is semi-symmetrical. For rooks the number is 8, 

 and a solution is obvious. I leave to any who are interested 

 in the subject the determination of the number of solutions in 

 each case. 



In connexion with this class of problems, I may mention 

 two other questions, to which Captain Turton first called my 

 attention, of a somewhat analogous character. 



The first of these is to place eight queens on a chess-board 

 so as to command the fewest possible squares. Thus, if queens 

 are placed on cells 21, 22, 62, 71, 73, 77, 82, 87, eleven cells on 

 the board will not be in check; the same number can be 

 obtained by other arrangements. Is it possible to place the 

 eight queens so as to leave more than eleven cells out of check? 

 I have never succeeded in doing so, nor in showing that it is 

 impossible to do it. 



The other problem is to place m queens (m being less than 

 5) on a chess-board so as to command as many cells as possible. 

 For instance, four queens can be placed in several ways on the 

 board so as to command 58 cells besides those on which the 

 queens stand, thus leaving only 2 cells which are not com- 

 manded; for instance, queens may be placed on the cells 35, 41, 



* The problem for knights waB discussed in L' IntermSdiaire des MatM- 

 maticiens, Paris, 1896, vol. ni, p. 58; 1897, vol. iv, pp. 15—17, 254; 1898, vol. 

 v, pp. 87, 230—231. 



