CH. VI] 



CHESS-BOARD RECREATIONS 



115 



The solutions which involve c^, e 6 and e, can be written down 

 by symmetry. The eight solutions thus obtained are all dis- 

 tinct ; we may call them of the first type. 



The above are the only solutions which can involve elements 

 in the corner squares of the determinant. Hence the remaining 

 solutions are obtainable from the determinant 



6 a c, d 4 



ft a, 6 4 c, d„ 

 7« ft a, b„ Oy 



84 7» ft °7 6 8 



8, 7, ft 



If, in this, we take the minor of 6 2 and in it replace by zero 

 every term involving the letter b or the suffix 2 we shall get 

 all solutions involving b 3 . But in this case the minor at once 

 reduces to d 6 a 6 S 4 ft. We thus get one solution, namely, & 2 d 6 a„&4j8 s . 

 The solutions which involve /8 a , S 4 , £,, /3 a , b a , d t , and d 4 can be 

 obtained by symmetry. Of these eight solutions it is easily 

 seen that only two are distinct : these may be called solutions 

 of the second type. 



Similarly the remaining solutions must be obtained from 

 the determinant 



c, 

 a, 64 c, 



7» ft 



b, Ct 



7« ft «r 

 7, 



If, in this, we take the minor of c„, and in it replace by zero 

 every term involving the letter c or the suffix 3, we shall get 

 all the solutions which involve c,. But in this case the minor 

 vanishes. Hence there is no solution involving c„ and therefore 

 by symmetry no solutions which involve 7 8 , 7,, or &,. Had there 

 been any solutions involving the third element in the first or 

 last row or column of the determinant we should have described 

 them as of the third type. 



8—2 



