1902.] on Magic Squares and other Problems on a Chess-Board. 59 



was one of its first victims, a solution of a more elegant nature being 

 obtained tban tbat whicb bad resulted from the first j)lan of operations. 

 There is such an extensive choice of processes and functions that 

 many solutions are obtainable of any particular problem. I will now 

 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 most suitable to 

 explain to an audience. Sujipose we have five collections of objects, 

 each collection containing the same five different objects, a, h, c, d, e 

 (Fig. 12). I suppose the objects distributed amongst five different 



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



(.bcde) (a.cde) {ab.de') (abc.e) {cibcd.) 



(. .cde) (a.c.e) (ab.d.) (^.bc.e) (ab.d.) 



l...de) (a.c.) (ab...) (..c.e) (.b,d.) 



(.....) (a....) (.&...) (.....) (...d.) 



Fig. 12. 



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, d 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 in- 

 tention of taking one object from each collection and obtaining 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, etc. However, he has a good choice in his selection, and we 

 will suppose him to take b from the first collection, d from the second, 

 p from the third, a from the fourth, c from the fifth. The collections 

 then become 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 c 

 from the first, e from the second, d from the third, b 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 selec- 

 tions tbat 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 pro- 

 cess 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 

 square are also solved. 



