60 



Major P. A. MacMahon 



[Feb. 14, 



I will now mention some questions of a more difficult character 

 that are readily solved by the method. In the " probleme des 

 menages " you will recollect that the condition w^as that no man must 

 sit next to his wife. If the condition be that there must be at least 

 four (or any even number) persons between him and his wife, the 



question is just as easily solved. Latin squares where the letters 

 are not all different in each row and column are easily counted. 

 Illustrations of these are shown in Fig. 13. One of these extended 

 to order 8 gives the solution of the problem of placing 16 castles on 

 a chess-board, 8 black and 8 white, so that no castle can take another 

 of its own colour. 



Theoretically, the G-raBCO-Latin squares of Euler can be counted, 

 but I am bound to say that the most laborious calculations are neces- 

 sary to arrive at a numerical result, or even to establish that in 

 certain cases the number sought is zero. 



Fig. 14. 



Next consider a square of any size and any number of different 

 letters, each of which must appear iu each row and in each column, 

 while there is no restriction as to the number that may appear in any 

 one compartment. In this case the result is very simple: suppose 

 the square of order 4 (Fig. 14), and that there are seven ditierent 



