114 OHESS-BOAKD EECREATIONS [CH. VI 



same suffixes lie along bishop's paths : hence, if we retain only 

 those terms in each of which all the letters and all the suffixes 

 are different, they will denote positions in which the queens 

 cannot take one another by moves bishop-wise. It is clear that 

 the signs of the terms are immaterial. 



In the case of an ordinary chess-board the determinant is of 

 the 8th order, and therefore contains 8 !, that is, 40320 terms, 

 so that it would be out of the question to use this method 

 for the usual chess-board of 64 cells or for a board of larger 

 size unless some way of picking out the required terms could 

 be discovered. 



A way of effecting this was suggested by Dr J. W.L. Glaisher* 

 in 1874, and so far as I am aware the theory remains as he left 

 it. He showed that if all the solutions of n queens on a board 

 of w s cells were known, then all the solutions of a certain type 

 for n + 1 queens on a board of (to + l) 2 cells could be deduced, 

 and that all the other solutions of n + 1 queens on a board of 

 (n + 1) 8 cells could be obtained without difficulty. The method 

 will be sufficiently illustrated by one instance of its application. 



It is easily seen that there are no solutions when n = 2 and 

 n= 3. If w = 4 there are two terms in the determinant which 

 give solutions, namely, & s (vy,/8 6 and cS^yt- To find the solutions 

 when n = 5, Glaisher proceeded thus. In this case, Giinther's 

 determinant is 



<h b 2 c» d t e s 



A a>» b t c t d„ 



7, & a, &„ c, 



K 7» A a, b a 



e« S 6 7, £ 8 a, 



To obtain those solutions (if any) which involve a, it is sufficient 

 to append a, to such of the solutions for a board of 16 cells as 

 do not involve a. As neither of those given above involves an 

 a we thus get two solutions, namely, bjCiy^a, and c 3 /3 a b 6 y,,a 9 . 



* Philosophical Magazine, London, December, 1874, aeries 4, vol. xlvui 

 pp. 457—467. 



