Mr. J. W. L. Glaisher on the Problem of the Eight Queens. 461 



reflexion k 5 c 4 d 3 e 9 b 8 h 7 and h 5 b 4 e 3 d 9 c 8 k 7 ; and as there are no ante- 

 penultimate solutions, these four penultimate solutions are all 

 that the problem admits of. 



Consider now a board of 49 squares, 



d 3 b 4 



e 3 9 A 



n 5 



9g 



e* 



m, 



a 5 



d 7 



n 6 ?7 



k 7 n 8 



98 ^9 



e 9 9\o 



C\r\ en 



d n b 



p 7 m 8 he 



'9 

 flO 



10 



d n b ]2 



'\3 



(3) 



As not one of the 4 six-solutions involves an a, we obtain by 

 merely appending a X3 to them the 4 seven-solutions that involve 

 « 1Q , viz. 



135 



k b c 4 d 3 e 9 b 8 h 7 a ]3 



b 9TbJ 89 4 e 7 C \0 a \3 



h 5 b 4 e 3 d 9 c 8 k 7 a l3 . 



V9 b l094^f6P7 

 e u98h<?9 b 6 a 3P7°> 



These by reflexion give 



n 6 e 5 b 49i0^f8P- 



e 3 96 k 9 d s b » a nP7 



and the eight corresponding ultimate solutions that involve the 

 other corner squares «, and q 7 are written down at once by re- 

 placing each suffix by its complement to 14 in the first four, 

 and by interchanging b and c, d and e, f and g } h and k t m and 

 n, p and q in the last four. 



On developing the determinants in the manner explained 

 above, it appears that there are no antepenultimate or preante- 

 penultimate solutions, and that the penultimate solutions which 

 involve c 2 are six in number, viz. 



Cc,n 8 e 7 b 6 h 5 a n f lQt (i) 



c<2k 7 a 59io b 8 m 6 d n> • (ii) 



c * e s d a9\<P9f* b \*> (iii) 



c*9G a sf4 e \\ d s h \v • ( iv ) 



C296 1( 9 a 7 h bf8 b \V (V) 



c 29e d 3 a 7 e \\f8 b \<2 (vi) 



The vice squares are n 8 , b }<2) m 6 ; and on turning the board 

 through 90° so that n 8 occupies the position of the leading 

 square c 2 , we obtain five new solutions : 



