44 



THE MAGIC SQUARE. 



or 15, 8, 2, 9, or 14, 12, 3, 5. We may easily convince ourselves 

 that this square is obtainable from the square of Diirer by inter- 

 changing with one another the two middle vertical rows. 



ii. 



EARLY METHODS FOR THE CONSTRUCTION OF ODD-NUMBERED 



SQUARES. 



Since early times rules have also been known for the construc- 

 tion of magic squares of more than 3 times 3, or 4 times 4 spaces. 

 In the first place, it is easy to calculate the sum which in the case 

 of any given number of cells must result from the addition of each 

 row. We take the determinate number of cells in each side of the 

 square which we have to fill, multiply that number by itself, add i, 

 again multiply the number thus obtained by the number of the cells 

 in each side, and, finally, divide the product by 2. Thus, with 4 

 times 4 cells or squares, we get : 4 times 4 are 16, 16 and i are 17, 

 'and one half of 17 times 4 is 34. Similarly, with 5 times 5 squares, 

 we get : 5 times 5 are 25, and i makes 26, and the half of 26 times 

 5 is 65. Analogously, for 6 times 6 squares the summation in is 

 obtained, for 7 times 7 squares 175, for 8 times 8 squares 260, for 9 

 times 9 squares 369, for 10 times 10 squares 505, and so on. The 

 Hindu rule for the construction of magic squares whose roots are 

 odd, may be enunciated as follows : To start with, write i in the 

 centre of the topmost row, then write 2 in the lowest space of the 



i?5 *75 175 175 175 175 175 

 Fig. 5- 



vertical column next adjacent to the right, and then so inscribe the 

 remaining numbers in their natural order in the squares diagonally 

 upwards towards the right, that on reaching the right-hand margin 





