CH. VII] 



MAGIC SQUARES 



141 



cells in C. Of course the resulting square remains magic in 

 columns. It will also now be magic in rows and diagonals, since 

 the construction is equivalent to writing in each of the quarters 

 A, B, C, D, equal magic squares of the order u made with 

 the numbers 1 to w 2 , and then superposing on them a magic 

 square of the nth order made with the four radix numbers 0, v?, 

 2w 2 , 3m 2 , each repeated w a times. The component 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 n 1 appears once and only once in 

 the resulting square. 



Figures v and vi show the application of the rule to the 

 construction of a magic square of the sixth order. In figure v, 



Figure v. Initial Quarter-Squares. 



Figure vi. Final Square, n=6. 



Figure vii. Final Square, n=6. 

 those numbers in the cells in the initial quarter-squares A and 

 B which are to be interchanged vertically with the numbers in 

 the corresponding cells in D and G are underlined ; figure vi 

 represents the final square obtained ; and figure vii shows how 



