THE MAGIC SQUARE. 



43 



however that 5 necessarily must occupy the middle place, and that 

 the even numbers must stand in the corners. This being so, there 

 are but 7 additional arrangements possible, which differ from the 

 arrangement above given and from one another only in the respect 

 that the rows at the top, at the left, at the bottom, and at the right, 

 exchange places with one another and that in addition a mirror be 

 imagined present with each arrangement. So too from Durer's 

 square of 4 times 4 places, by transpositions, a whole set of new 

 correct squares may be formed. A magic square of the 4 times 4 

 numbers from i to 16 is formed in the simplest manner as follows. 

 We inscribe the numbers from i to 16 in their natural order in tJe 

 squares, thus : 



Fig. 3- 



We then leave the numbers in the four corner-squares, viz. i, 4, 13, 

 1 6, as well also as the numbers in the four middle-squares, viz. 6, 

 7, 10, u, in their original places ; and in the place of the remaining 

 eight numbers, we write the complements of the same with respect 

 to 17 : thus 15 instead of 2, 14 instead of 3, 12 instead of 5, 9 in- 

 stead of 8, 8 instead of 9, 5 instead of 12, 3 instead of 14, and 2 in- 

 stead of 15. We obtain thus the magic square 



from which the same sum 34 always results. It is an interesting 

 property of this square that any four numbers which form a rectangle 

 or square about the centre also always give the same sum 34 ; for 

 example, i, 4, 13, 16, or 6, 7, 10, u, or 15, 14, 3, 2, or 12, 9, 5, 8, 



