1902.J o)t Magic Squares and nfJicr Pro^>lem>^ on a Chcsi^-Bonrd. 5-i 



formation, but in the enumeration of all possible ways of forming 

 them, and in this respect very little has been hitherto achieved by 

 mathematicians. No person living knows in how many ways it is 

 possible to form a magic square of any order exceeding 4. The fact is, 

 that before we can attempt to enumerate magic squares we must see 

 our way to solve problems of a far more simple character. For ex- 

 ample, before we can enumerate the squares that can be formed by 

 De la Hire's method we must take a first step by finding out how 

 many Latin squares can be formed of the different orders. For 

 the order 5 the question is, " In how many ways can five different 

 objects be placed in the cells so that each column and each row 

 contains each object ? " It may occur to some here this evening that 



2_ 3_J_J__5 

 5 "2 3 TfT 



4 I 5 ra 



Fro. f). 



such a discussion might be interesting or curious, but could not pos- 

 sibly be of any scientific value. But such is not the case. A depart- 

 ment of mathematics that is universally acknowledged to be of funda- 

 mental importance is the " theory of groups." Operations of this theory 

 and those connected with logical and other algebras possess what is 

 termed a "multiplication table," which denotes the laws to which 

 the operations are subject. In Fig. 6 you see such a table of order 6 

 slightly modified from Burnside's ' Treatise on the Theory of 

 Groups ' ; it is, as you see, a Latin square, and the chief pro- 

 blem that awaits solution is the enumeration of such tables; the 

 questions are not parallel because all Latin squares do not give rise 



