THE MAGIC SQUARE. 



}f the numbers and must then find the complements of the numbers 

 with respect to some other certain number (as 17 in the square ot 

 f) and also effect certain exchanges of the numbers with one an- 

 other. To form, for example, a magic square of 6 times 6 places, 

 we inscribe in the 12 diagonal cells the numbers that in the natural 

 sequence of inscription fall into these places, then in the remaining 

 cells the complements of the numbers that belong therein with re- 

 spect to 37, and finally effect the following six exchanges, viz. of 

 the numbers 33 and 3, 25 and 7, 20 and 14, 18 and 13, 10 and 9, 

 and 5 and 2. In this way the following magic square is obtained. 



Fig. 15- 



This square may also be constructed by the method of De la 

 Hire, from two auxiliary squares with the numbers i, 2, 3, 4, 5, 6 

 and o, 6, 12, 18, 24, 30 respectively. In this case, however, the 

 vertical rows of the one square and the horizontal rows of the other 

 must each so contain two numbers three times repeated that the 

 summation shall always remain 21 and go respectively. In this 

 manner we get the magic square last given above from the two fol 

 lowing auxiliary squares : 



and 



Fig. 16. 



Fig. 17. 



It is to be noted in connection with this example that here also 

 as in the case of odd-numbered squares, it is possible so to inscribe 



