116 CHESS-BOARD RECREATIONS [CH. VI 



Thus in all there are ten and only ten solutions, namely, 

 eight of the first type, two of the second type, and none of the 

 third type. 



Similarly, if n = 6, we obtain no solutions of the first type, 

 four solutions of the second type, and no solutions of the third 

 type ; that is, four solutions in all. If n = 7, we obtain sixteen 

 solutions of the first type, twenty-four solutions of the second 

 type, no solutions of the third type, and no solutions of the 

 fourth type ; that is, forty solutions in all. If n = 8, we obtain 

 sixteen solutions of the first type, fifty-six solutions of the second 

 type, and twenty solutions of the third type, that is, ninety-two 

 solutions in all. 



It will be noticed that all the solutions of one type are not 

 always distinct. In general, from any solution seven others can 

 be obtained at once. Of these eight solutions, four consist of 

 the initial or fundamental solution and the three similar one.- 

 obtained by turning the board through one, two, or three right 

 angles ; the other four are the reflexions of these in a mirror . 

 but in any particular case it may happen that the reflexions 

 reproduce the originals, or that a rotation through one or two 

 right angles makes no difference. Thus on boards of 4 2 , 5 2 , 6 2 , 

 7 2 , 8 2 , 9 2 , 10 2 cells there are respectively 1, 2, 1, 6, 12, 46, 92 

 fundamental solutions ; while altogether there are respectively 

 2, 10, 4, 40, 92, 352, 724 solutions. 



The following collection of fundamental solutions may in- 

 terest the reader. Each position on the board of the queens 

 is indicated by a number, but as necessarily one queen is on 

 each column I can use a simpler notation than that explained 

 on page 109. In this case the first digit represents the number 

 of the cell occupied by the queen in the first column reckoned 

 from one end of the column, the second digit the number in the 

 second column, and so on. Thus on a board of 4* cells the 

 solution 3142 means that one queen is on the 3rd square of the 

 first column, one on the 1st square of the second column, one 

 on the 4th square of the third column, and one on the 2nd 

 square of the fourth column. If a fundamental solution gives 

 rise to only four solutions the number which indicates it is placed 



