137 



CHAPTER VII. 



MAGIC SQUARES. 



A Magic Square consists of a number of integers arranged 

 in the form of a square, so that the sum of the numbers in 

 every row, in every column, and in each diagonal is the same. 

 If the integers are the consecutive numbers from 1 to n 2 the 

 square is said to be of the nth order, and it is easily seen that 

 in this case the sum of the numbers in every row, column, and 

 diagonal is equal to \n (n 2 + 1) : this number may be denoted 

 by N. Unless otherwise stated, I confine my account to such 

 magic squares, that is, to squares formed with consecutive 

 integers from 1 upwards. The same rules cover similar pro- 

 blems with n 2 numbers in arithmetical progression. 



Thus the first 16 integers, arranged in either of the forms 

 in figures i and ii below, represent magic squares of the fourth 



ra=4. 



Figure i. n = i. Figure ii. 



order, the sum of the numbers in each row, column, and diagonal 

 being 34. Similarly figure iii on page 139, figure xiii on page 

 145, and figure xvi on page 151 represent magic squares of the 

 fifth order; figure vi on page 141 represents a magic square of 

 the sixth order; figure xviii on page 154 represents a magic 

 square of the seventh order ; figure xii on page 143 and figure 

 xxii on page 157 represent magic squares of the eighth order; 

 figure xxiii on page 157 represents a magic square of the ninth 



