CH. VII] 



MAGIC SQUARES 



139 



4m. In each case, I now give the simplest rule with which I am 

 acquainted, omitting alternative methods described in previous 

 editions of this work. 



Magic Squares of an Odd Order. A magic square of the 

 «th order, where n = 2m + 1, can be constructed by the following 

 rule due to De la Loubere*. First, the number 1 is placed in 

 the middle cell of the top row. The successive numbers are 

 then placed in their natural order in a diagonal line which 

 slopes upwards to the right, except that (i) when the top row 

 is reached the next number is written in the bottom row as if 

 it came immediately above the top row ; (ii) when the right- 

 hand column is reached, the next number is written in the 

 left-hand column, as if it immediately succeeded the right-hand 

 column ; and (iii) when a cell which has been filled up already, 

 or when the top right-hand square is reached, the path of the 

 series drops to the row vertically below it and then continues 

 to mount again. Probably a glance at the diagram in figure iii, 

 showing the construction by this rule of a square of the fifth 

 order, will make the rule clear. 



Figure iii. n = 5. 



Figure iv. n=5. 



The reason why such a square is magic can be best explained 



by taking a particular case, for instance, n = 5, and expressing 



all the numbers in the scale of notation whose radix is 5 (or n, 



if the magic square is of the order n), except that 5 is allowed 



to appear as a unit-digit and is not allowed to appear as a 



unit-digit. The result is shown in figure iv. From that figure 



it will be seen that the method of construction ensures that 



* S. De la Loubere, Dv Boyaume de Siam (Eng. Trans.) , London, 1693, vol. n, 

 pp. 227 — 247. De la Loubere was the envoy of Louis XIV to Siam in 1687-8, 

 and there learnt this method. 



