CH. VII] 



MAGIC SQUARES 



159 



cells, of r? counters (the counters being divided into n groups, 

 each group consisting of n counters of the same colour and 

 numbered consecutively 1,2,..., ri), so that each row and each 

 column shall contain no two counters of the same colour or 

 marked with the same number. Such arrangements are termed 

 Eulerian Squares. 



For instance, if n = 3, with three red counters Oj, a 2 , a s , 

 three white counters b lt 6 2 , b d , and three black counters c u c 2 , c 8 , 

 we can satisfy the conditions by arranging them as in figure xxiv 

 below. If n = 4, then with counters a^, a 2 , a s , a t ; b u b 2 , b s , 6 4 ; 

 Cj, c 2 , c 8 , c 4 ; d lt d 2 , d s , d„ we can arrange them as in figure xxv 

 below. A solution when n = 5 is shown in figure xxvi. 



Figure xxiv. 



Figure xxv. 



Figure xxvi. 



The problem is soluble if w is odd; it would seem* that it is 

 insoluble if n is of the form 2 (2m + 1). If solutions when n = a 

 and when n = b are known, a solution when n — ah can be written 

 down at once. If n is an odd prime greater than 3, a solution 

 can be given which covers the diagonals as well as the rows and 

 columns of the square. The theory is closely connected with 

 that of magic squares. 



Magic Domino Squares. Magic problems can be made with 

 dominoes. An ordinary set of dominoes, ranging from double 

 zero to double six, contains 28 dominoes. Each domino is a 

 rectangle formed by fixing two small square blocks together 

 side by side : of these 56 blocks, eight are blank, on each of 

 eight of them is one pip, on each of another eight of them are 

 two pips, and so on. It is required to arrange the dominoes so 

 that the 56 blocks form a square of 7 by 7 bordered by one line 

 * C. Planck, The Monist, Chicago, vol. xxix, 1919, p. 308. 



