CH. VII] 



MAGIO SQUARES 



143 



lined, while figure ix represents the final magic square of the 

 tenth order thus obtained. 



Magic Squares of a Doubly-Even Order. A magic 

 square of the nth order, where n = 4m, can be constructed by 

 the following rule*. Begin by filling the cells of the square 

 with the numbers 1, 2, ..., n 2 , written in their natural order 

 from left to right and taking the rows in succession from the 

 top. Divide the square into four equal quarters A, B, G, D, as 

 represented in figure x. Divide each of these quarter-squares 

 again into four equal parts, each of which will contain m? cells, 

 as shown in figure xi. Then if the numbers in the cells a^ and 

 a 4 are interchanged with those in the skewly related cells in C, 



Figure x. 



Figure xi. 



Figure xii. A Magic Square, n=8. 



and the numbers in the cells b 2 and b 3 are interchanged with 



those in the skewly related cells in B, the resulting square will 



be magic. A magic square of the eighth order constructed by 



this rule is shown in figure xii. 



* The rule seems to have been first enunciated in 1889 by W. Firth, but 

 later was independently discovered by various writers: see the Messenger of 

 Mathematics, Cambridge, September, 1893, vol. xxm, pp. 65 — 69, and the 

 Monist, Chicago, 1912, vol. xxn, pp. 53 — 81. 



