1902.] on Magic Squares and otliPr Problems on a, Chess-Board. 61 



letters tliat must appear in each row and column ; the number of ar- 

 rangements is (4 !)'^, viz., 4 the order of the square, must be multi- 

 plied 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 are 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 diffi- 

 cult when the lattice is rectangular instead of square and when 

 any desired limitation 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 different, arises 

 from the processes of 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 diffi- 

 culty is involved in the condition that no two numbers must be the 

 same. Still, a statement can be made as to a succession of mathe- 

 matical 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 mathematics that it has 

 been fascinating to forge and to strengthen. Of all the subjects that 

 for hundreds of years past have from time to time engaged the atten- 

 tion 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 



