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



There were 4 seven-solutions that involved no a; and these 

 give rise to the following 4 solutions involving a [b : 

 ai b n % e 9 k b f lQ d 7 b A m 6 o lb m^ 9 h 6 g l0 e 7 c A v 6 



a \S V 8 C ]0 e 7ff4 h 9 d 5 m 6 a \b m S b \0 d 7h h e b n 6> 



which give on rotation, 



r 8 a 13 b \09m d 5 C 6 k 7 e 3 r 8 a 3 b 69A \ C \0 K e \B 

 r 8 a 13 d 9 C \0 k l Aff6 e 3 r 8 a 3 d 7 C 6 k b b l2ffl0 e \3 5 



and by replacing every suffix by its complement to 16 in the 

 first four, and every letter by its conjugate in the second four, 

 we obtain the other 8 solutions involving a l and s Q . There are 

 thus 16 ultimate solutions. 



On expanding the determinants there are found to be 8 

 penultimate solutions involving c 2 , viz. 



Cie h g 8 k xx d 7 m^a x3 f 2 



c 2 ff6 k 9f4d 7 e 13 b 12 h n 



c 2 g 6 k 9 a 7 h 5 e l3 b 12 m lQ 



C 2 K 7 a 3 gi O 8 ei 3«g/ 1 2 



c 2 k 7 n 10 OQh 5 d 9 a 13 ji 2 

 c 2 n 8 e 7 k n h b d 9 b 12 m 10 

 c 2 n 8 a b e 9 g 12 b ]0 p 7 h n 



c 2Q9y8A d 7°io a i3hn> 

 six antepenultimate solutions involving e 3) viz. 

 e 3 g&b 4 k n h b c l2 f^d X3 

 e 3 gQn J0 a 7 h 5 c l2 m 8 d 13 

 c 3 n 8 b 4 k n a 9 m 6 f 10 d 13 

 e 3 k 7 b 4 c 8 g l2 m^a l3 h n 

 ezk 7 n lQ f 4 a 9 c X2 m 8 h n 

 e 3 k 7 n l0 d b b 8 m 6 a 13 f l2 , 



and no preantepenultimate solutions. 



Considering first the penultimate solutions, we see that the 

 first six are unsymmetrical and the last two quasi-symmetrical ; 

 the former give rise by rotation to 18 additional solutions: 



qsgioanfuc^mjis b 14 d n f 8 h b e 9 n 10 a 3 g 4 p 7 f(ia b g 4 b 10 d 13 n 8 k n 



q 9 e n bi 2 k b c 6 m 10 h 7 f 4 b 14 f 1& h 7 g 12 e 9 d 3 c 4 k b p 7 d b c 4 h u b lQ n 6 k 9 g 12 



q 9 ei ib l2 c 8 g 4 m 10 h 7 d 3 b 14 f 10 h 7 a 9 k n d 3 c 4 n 6 p 7 d b c 4 b 8 f l2 n 6 k9e )3 



q 9 c 12 n 6 d n a 7 m 10 b 4 h b b 14 h 9 e l3 f 6 c 8 d 3 k 7 g 4 p 7 b 4 m 1Q e b a 9 n 6 c x2 k n 



q 9 C\%d x3 e 7 g 4 btfn 8 h h b 14 h 9 m 6 c 10 k u e 7 a 3 g 4 p 7 b 4 e 3 d g f l2 c lQ n 8 k n 



^9«i3Cio/i2^AM3 b u m 8 d 9 n 5 k u e 7 c 4 n 6 p 7 a 3 b 6 g 4 / 12 c l0 k 9 e 13 ; 



