54 



THE MAGIC SQUARE. 



sum is divisible by 3, magic squares will always be obtainable with 

 3 times 3 spaces which shall give this sum. In such a case we di- 

 vide the sum required by 3 and subtract 5 from the result in order 

 to obtain the number which we have to add to each number of the 

 original square. If the sum desired is even but not divisible by 4, 

 we must then subtract from it 34 and take one fourth of the result, 

 to obtain the number which in this case is to be added in each 

 place. If, for example, we wish to obtain the number of the year 

 1890 as the resulting sum of each row, we shall have to add to each 

 of the numbers of an ordinary magic square of 4 times 4 spaces the 

 number 464 ; in other words, instead of the numbers from i to 16 we 

 have to insert in the squares the numbers from 465 to 480. As the 

 number of the year 1892 is divisible by eleven, it must be pos- 

 sible to deduce from the magic square constructed by us at the con- 

 clusion of Section III a second magic square in which each row of 

 TJ cells will give the number of the year 1892. To do this, we sub 

 tract from 1892 the sum of the original square, namely 671, and di- 

 vide the remainder by n, whereby we get in and thus perceive 

 that the numbers from 112 to 232 are to be inscribed ir the cells of 



1892 1892 1892 1892 1892 1892 1892 1892 1892 1892 1892 

 Fig. 24. 



the square required. We get in this way the preceding square, from 

 which one and the same sum, namely 1892, can be obtained 44 times, 

 first from each of the 1 1 horizontal rows, secondly from each of the 

 ii vertical rows, thirdly from each of the two diagonal rows, and 



