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



ri s b ^ a 9 c 6 e 3 n 7f^ ...... (vii.) 



*tfifof\&b& l .e> ( viii ) 



n S C ^ e 79A h 9 d h m 6> ....... (1X) 



n 8 c l0 d n a 7 ej) 4 m 6 , ....... (x) 



n 8 c ]Q k 5 a 7 h 9 b 4 m 6 (xi) 



(ii) gives no new solution, as n 8 a n e 7 f l0 b G c^h 6 which results from 

 it is only a reproduction of (i). A further rotation through 90°, 

 so that b 12 becomes the leading square, gives only one new so- 

 lution, 



b 12 m b ^ 7 c 8 k 9 a a ff A ' y (xii) 



and another rotation through 90° gives no new solution. By 

 reflecting (i) . . . (xii) we obtain 12 more solutions, so that for 

 the board of 49 squares there are 40 solutions, of which 16 are 

 ultimate and 24 penultimate. 



To rotate a solution through 90° in the simplest manner, 

 observe that the corresponding letters on opposite sides of the 

 a diagonal form the pairs b and c, d and e, f and g, h and k, m 

 and n } p and q, so that in any solution to replace each letter by its 

 fellow in its pair (or, say, by its conjugate) is equivalent to turn- 

 ing the board through 90° and then reflecting the result. Thus 

 the process is to (mentally) replace each letter by its conjugate; 

 and (keeping the eye on the representation of the board) to reflect 

 each square, e. g, to rotate (i) through 90°, we take from (3) 

 the reflection of b 2 , viz. n 8 ; of m 8 , viz. Z> 12 ; of d 7 , viz. a 9 , and so 

 on, thus obtaining (vii). The operations are very readily per- 

 formed; and it is convenient to always reflect in the manner 

 doted above, viz. by replacing the rth square from one side by 

 the rth square from the other side in the same row. The rota- 

 tion through 180° is effected by merely replacing each letter by 

 its conjugate, and each suffix by its complement (as in xii, which 

 is so derived from i) ; and the rotation through 270° is effected by 

 treating similarly the 90° results. Thus (i) turned through 

 270° gives m 6 c^a 5 b 8 d n k 7 g 10 , which is identical with (ii). It may 

 be noticed that we always obtain a verification of the result 

 thus : — As (^(vi) are all the solutions in which c 2 is involved, 

 we see that the vice squares n 8 , & 12 , m 6 can each only be involved 

 in 6 of the solutions. On counting the number of solutions in 

 which they occur in (i)-(xii), it will be seen that this is the 

 case ; and the most certain way of performing the work is to 

 write down all the solutions that result from the rotations 

 through 90°, 180°, and 270°, strike out those that have appeared 

 previously, and then see that in the solutions which remain the 

 leading square and the three vice squares each appear the same 

 number of times. 



