156 MAGIC SQUARES [CH. VII 



10, 30, 1, 28, 48, 19, 39 form a magic group connected cyclically 

 by steps (2, 3). 



Symmetrical Squares. It has been suggested that we might 

 impose on the construction of a magic square of the order n the 

 condition that the sum of any two numbers in cells skewly 

 related to one another shall be constant and equal to m 2 + 1. 

 Such squares are called Symmetrical. 



The construction of odd symmetrical squares of the order n, 

 when n is prime to 3 and 5, involves no difficulty. We can 

 begin by placing the mean number in the middle cell and work 

 from that, either in both directions or forwards, making the 

 number 1 follow after n a ; we can also effect the same result 

 by constructing a pandiagonal square of the order n and then 

 transposing a certain number of rows and columns. If the rule 

 given above on page 154, where a = 1, b = 2, h = 2, k = — 1, be 

 followed, this will lead to placing the number 1 in the (ra+3)/2th 

 cell of the top row, as is exemplified in figure xviii. 



Such a square must be symmetrical, for if we begin with 

 the middle number (w 2 + 1)/2, which I will denote by m, in the 

 middle cell, and work from it forwards with the numbers m + 1, 

 m + 2, ..., and backwards with the numbers m — 1, m — 2, ..., 

 the pairs of cells filled by the numbers m + 1 and m — 1, to + 2 

 and m — 2, &c, are necessarily skewly related, and the sum of 

 the numbers in each pair is 2m. This was first pointed out by 

 McClintock. 



The construction of doubly-even symmetrical pandiagonal 

 squares is also possible, but the analysis is too lengthy for me 

 to find room here for it. 



In a symmetrical square any n such pairs of numbers 

 together with the number in the middle cell will form a magic 

 group. For instance in figure xviii, the group 32, 18, 36, 14, 47, 3, 

 25 is magic. So also is the group 47, 3, 35, 15, 13, 37, 25. Thus 

 in a symmetrical pandiagonal square, even of a comparatively 

 low order, there are hundreds of magic groups of n numbers 

 whose sum is constant. 



Doubly-Magic Squares. In another species of hyper-magic 

 squares the problem is to construct a magic square of the nth 



