140 MAGIC SQUARES [CH. VII 



every row and every column shall contain one and only one of 

 each of the unit-digits 1, 2, 3, 4, 5, the sum of which is 15; and 

 also one and only one of each of the radix-digits 0, 5, 10, 15, 20, 

 the sum of which is 50. Hence, as far as rows and columns are 

 concerned, the square is magic. Moreover if the square is odd, 

 each of the diagonals will contain one and only one of each of 

 the unit-digits 1, 2, 3, 4, 5. Also the leading diagonal will 

 contain one and only one of the radix-digits 0, 5, 10, 15, 20, the 

 sum of which is 50 ; and if, as is the case in the square drawn 

 above, the number 10 is the radix-digit to be added to the 

 unit-digits in the other diagonal, then the sum of the radix- 

 digits in that diagonal is also 50. Hence the two diagonals also 

 possess the magical property. 



The effect of the construction is that we superpose two 

 magic squares, one made with the unit-digits 1, 2, 3, 4, 5, each 

 repeated five times, and the other with the radix-digits 0, 5, 10, 

 15, 20, each repeated five times. These squares being magic, 

 the square resulting from their superposition must be magic, 

 and they are so formed that their superposition ensures that 

 every number from 1 to 25 appears once and only once in the 

 resulting square. 



Magic Squares of a Singly-Even Order. A magic square 

 of the nth. order, where n = 2 (2m + 1), can be constructed by 

 the following rule due to Ralph Strachey*. Divide the square 

 into four equal quarters A, B, G, D, as shown in figure x on 

 page 143. Construct in A, by De la Loubere's method, a magic 

 square with the numbers 1 to w 2 where u = n/2. Construct by 

 the same rule, in B, G, D, similar magic squares with the 

 numbers w 2 + 1 to 2m 2 , 2m 2 + 1 to 3m 2 , and 3« 2 + 1 to 4u 2 . Clearly 

 the resulting composite square is magic in columns. In the 

 middle row of A take the m cells next but one to the left-hand 

 side, and in each of the other rows take the m cells nearest to 

 the left-hand side, interchange the numbers in these cells with 

 the numbers in the corresponding cells in B. Next interchange 

 the numbers in the cells in each of the m + 2 columns next to 

 the left-hand side of B with the numbers in the corresponding 

 * Communicated to me in a letter, August 1918. 



