1902.] on Magic Squares and other Problems on a Chess-Board. 55 



the six regiments by the Latin letters a, h, c, d, e, /, and the six 

 ranks by the Greek letters a, (3, y, 8, e, 0, and observes that the 

 character of an officer is determined by a combination of two 

 letters, the one Latin and the other Greek; there are 36 such 

 combinations, and the problem consists in placing these combi- 

 nations in the 36 compartments in such wise that every row and 

 every column contains the 6 Latin letters and also the 6 Greek 

 letters (Fig. 7). Euler found no solution of this problem in the 

 case of a square of order 6, and since Euler's time no one has suc- 

 ceeded either in finding a solution or in j^roving that no solution 

 exists. Anyone interested, has, therefore, this question before him 

 at the present moment, and I recommend it to anyone present who 

 desires an exercise of his wits and a trial of his patience and inge- 

 nuity. It is easy to prove that when the square is of order 2, 

 viz. the case of 4 officers of two different ranks drawn from two dif- 

 ferent regiments, there is no solution ; Euler gave his opinion to the 

 effect that no solution is possible whenever the order of the square is 

 two greater than a multiple of four. In other simple cases he ob- 

 tained solutions ; for example, for the order 3, the problem of 9 

 officers of three different ranks drawn from three 

 different regiments, it is easy to discover the solution 

 shown in the diagram (Fig. 8), and, as demonstrated 

 by Euler, whenever one solution has been constructed 

 there is a simple process by which a certain number 

 of others can he derived from it. Now, if you look 

 at that diagram and suppose the Greek letters oblite- 

 rated, you will see that the Latin letters are arranged 

 so that each of the letters occurs in each row and in Fig. 8. 



each column, the magical jn'operty mentioned above, 

 and for this reason Euler termed such arrangements Latin squares and 

 stated that the first step in the solution of the problem is to enume- 

 rate the Latin squares of a given order. As showing the intimate 

 connection between the Grseco-Latin square of Euler and ordinary 

 magic squares, it should be noticed that the method of De la Hire, 

 by employing Latin and Greek letters for the elements in his two 

 subsidiary Latin squares, gives rise immediately to the Graeco-Latin 

 square of Euler. Euler says in regard to the problem of the Latin 

 square, " The complete enumeration of the Latin squares of a given 

 order is a very important question, but seems to me of extreme diffi- 

 culty, the more so as all known methods of the doctrine of combina- 

 tions appear to give us no help," and again, " the enumeration appears 

 to be beyond the bounds of possibility when the order exceeds 5." 

 Moreover, Cayley, in 1890, that is 108 years later, gave a resume of 

 what had been done in the matter, but did not see his way to a 

 solution of the question. Under these circumstances, you will see 

 how futile it is to expect a solution of the magic-square problem when 

 the far simpler question of the Latin square has for so long proved 

 such a tough nut to crack. The problem of the Latin square has 



