March 13, 1902] 



NA TURE 



449 



questions are not parallel because all Latin squares do not give 

 rise to tables in the theory of groups ; but still, we must walk 

 before we can run, and a step in the right direction is the 

 enumeration of all Latin squares. When I call to mind that 

 the theory of groups has an important bearing upon many 

 branches of physical science, notably upon dynamics, I consider 

 that I have made good my point. 



I now concentrate attention on these Latin squares, and ob- 

 serve that the theory of the enumeration has nothing to do with 

 the particular numbers that occupy the compartments ; the only 

 essential is that the numbers shall be different one from 

 another. My attention was first called to the subject of the 

 Latin square by a work of the renowned mathematician Euler, 

 written in 1782, entitled " Recherches sur une nouvelle espcce 

 de Quarrcs Magiques." I may say that Euler seems to have 

 been the first to grasp the necessity of considering squares 

 possessing what may be termed a magical property of a far less 

 recondite character than that possessed by the magic squares of 

 the ancients, and, as we shall see presently, he might have gone a 



step further in the same direction with advantage and have com- 

 menced with arrangements of a more simple character than that 

 of the Latin square, with arrangements, in fact, which present 

 no difficulties of enumeration, but which supply the key to the 

 unlocking of the secrets of which we are in search. He com- 

 mences by remarking that a curious problem had been exercising 

 the wits of many persons. lie describes it as follows : — There 



are 36 oflicers of six different ranks drawn from six different 

 regiments, and the problem is to arrange them in a square of 

 order 6, one officer in each compartment, in such wise that in 

 each row, as well as in each column, there appears an officer of 

 each rank and also an officer of each regiment. Of a single 

 regiment we have, suppose, a colonel, lieutenant-colonel, major, 

 captain, first lieutenant and second lieutenant, and similarly for 

 five other regiments, so that there are in all 36 officers who 

 must be so placed that in each row and in each column each 

 rank is represented, and also each regiment. Eulerdenotes the six 

 regiments by the Latin letters a, h, c, d, c,f, and the six ranks by 

 the Greek letters a, j3, 7, 5. e, B, and observes that the character 

 of an officer is determined by a combination of two letters, the 

 one Latin and the other (Ireek ; there are 36 such combina- 

 tions, and the problem consists in placing these combinations 

 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 



NO. 1689, VOL 65] 



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

 succeeded either in finding a solution or in proving that no 

 solution exists. Anyone interested has, therefore, this question 

 before him at the present moment, and I recommend it to any- 



one present who desires an exercise of his wits and a trial of 

 his patience and ingenuity. 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 different 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 obtained solutions ; for ex- 

 ample, 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 be 

 derived from it. Now if you look at that diagram and suppose 

 the Greek letters obliterated, you will see that the Latin letters 

 are arranged so that each of the letters occurs in each row and 

 in each column, the magical property 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 

 enumerate the Latin squares of a given order. As showing the 

 intimate connection between the 'Gr.ijco- 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 imme- 

 diately to the Grreco-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 difficulty, the 

 more so as all known methods of the doctrine of combinations 

 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 

 rhitmd of what had been done in the matter, but did not see 

 his way to a solution of the question. Under these circum- 

 stances, 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 eventually been com- 

 pletely solved, and in order to lead you up gradually to an 

 understanding of the method that has proved successful, I ask 

 you to look at the Latin square of order 5 that you see in the 

 diagram (Fig. 9). The first row of letters can be written 

 in any order, but not so the second row, for each column 

 when the second row is written must contain two different 



letters. We must, therefore, be able to solve the compara- 

 tively simple question of the number of possible arrange- 

 ments of the first two rows. For a given order of the 

 letters in the first row, in how many ways can we write the 



