March 13, 1902] 



NA TURE 



451 



give you an idea of a solution of the Latin square, which is not 

 the most elegant that has been found, but which is the mosl 

 suitable to explain to an audience. Suppose we have five 

 collections of objects, each collection containing the same five 

 different objects, a, b, c, d, e {Fig. 12). I suppose the objects 

 distributed amongst five different persons in the following 

 manner : — The first person takes one object from each collection, 

 so as to obtain each of the five objects ; he can do this in 120 

 different ways ; we will suppose that he takes a from the first, 

 b from the second, c from the third, rf from the fourth, e from 

 the fifth ; the collections then become as you see in Fig. 12, 

 second row. Now suppose the second man to advance with 

 the intention of taking one object from each collection and ob- 

 taining each of the five objects, he has not the same liberty 

 of choice as had the first, because he cannot take a from the 

 first collection or b from the second, &c. However, he has 

 a good choice in his selection, and we will suppose him to take 

 b from the first collection, rf from the second, e from the third, 



(ahcde) {abode) (abcde) (abode) (abode) 



a from the fourth, i' from the fifth. The collections then be 

 come as you see in the third row. The third man who has 

 the same task finds his choice more restricted, but he 

 elects to take r from the first, e from the second, d from 

 the third, * from the fourth and a from the fifth. The 

 fourth man finds he can take d, c, a, e, b, and this leaves e, a, 

 b, c, d for the last man. If we plot the selections that have 

 been made by the five men, we find the Latin square shown in 

 Fig. 9. 



Every division of the objects that can be made on this plan 

 gives rise to a Latin square, and all possible distributions give 

 rise to the whole of the Latin squares. Now it happens that 

 a mathematical process exists (connected with algebraical 

 symmetric functions) that acts towards a function representing 

 the five collections in exactly the same way as I have supposed 

 the men to act, and when the process is performed five times in 

 succession, an integer results which denotes exactly the number 

 of Latin squares of order 5 that can be constructed. Moreover, 

 en route the " probleme des rencontres" and the problems 

 connected with any definite number of rows of the space are 

 also solved. 



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 

 was 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 S 

 gives the solution of the problem of placing 16 castles on a 

 chessboard, 8 black and 8 white, so that no castle can take 

 another of its own colour. 



Theoretically, the GrK;co- Latin squares of Euler can be 

 counted, but I am bound lo say that the most laborious calcula- 

 tions are necessary to arrive at a numerical result or even to 

 establish that in certain cases the number sought is zero. 



Next consider a square of any size and any number of dif- 

 ferent letters, each of which must appear in 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 



NO. 1689, VOL. 65] 



that there are seven different letters that must appear in each 

 row and column ; the number of arrangements is (4 !)', viz., 

 4, the order of the square, must be multiplied by each lower 

 number and the number thus reached multiplied seven times by 

 itself. 



Finally, if there be given for each row and for each column 

 a different assemblage of letters and no restriction be placed 

 upon the contents of any compartment, the number of squares 

 in which all these conditions ar^satisfied can be counted. This, 

 of course, is a far more recondite question than that of the Latin 

 square, and cannot be attacked at all by any other method. 



I now pass to certain purely numerical problems. Suppose 

 we have a square lattice of any size and are told that 

 numbers are to be placed in the compartments in such wise that 

 the sums of the numbers in the different rows and columns are 

 to have any given values the same or different. This very 

 general question, hitherto regarded as unassailable, is solved 

 quite easily. The solution is not more difficult when the lattice 

 is rectangular instead of square and when any desired limita- 

 tion is imposed upon the magnitude of the numbers. 



Up to this point, the solutions obtained depend upon processes 

 of the differential calculus. A whole series of other problems, 

 similar in general character, but in one respect essentially dif- 

 ferent, arises from the processesof the calculus of finite differences. 

 Into these time does not permit me to enter. In the case of 

 magic squares as generally understood, the method brought 

 forward marks a distinct advance in connection with De la 

 Hire's method of formation by means of a pair of Latin squares, 

 but apart from this a great difficulty is involved in the condition 



that no two numbers must be the same. Still, a statement can 

 be made as to a succession of mathematical processes which 

 result in a number which enumerates the magic squares of a 

 given order. In any cases except those of the first few orders, 

 the processes involve an absolutely prohibitive amount of labour, 

 so that it cannot yet be said that a practical solution of the 

 question has been obtained. 



Scientifically speaking, it is the assignment of the processes 

 and not the actual performance of them that is interesting ; it is 

 the method involved rather than the results flowing from the 

 method that is attractive ; it is the connecting link between 

 two, to all appearance, widely separated departments of mathe- 

 matics that it has been fascinating to forge and to strengthen. 

 Of all the subjects that for hundreds of years past have from 

 time 10 time engaged the attention of mathematicians, perhaps 

 the most isolated has been the subject of these chess-board 

 arrangements. This isolation does not, I believe, any longer 

 exist. The whole series of diagrams formed according to any 

 given laws must be regarded as a pictorial representation, in 

 greatest detail, of the manner in which a certain process is 

 performed. We have to exercise our wits to discover what this 

 process is. To say and to establish that problems of the general 

 nature of the magic square are intimately connected with the 

 infinitesimal calculus and the calculus of finite differences is to 

 sum the matter up. Much, however, remains to be done. The 

 present method is not able to deal with diagonal properties, 

 or with arrangements which depend upon the knight's move. 

 The subject is only in its infancy at present. More workers 



