CH. VII] 



MAGIC SQUARES 



145 



3rd order we can build up successively squares of the orders 

 5, 7, 9, etc., that is, every odd magic square. Similarly from 

 a magic square of the 4th order we can build up successively 

 all higher even magic squares. 



The method of construction will be clear if I explain how 

 the square in figure xiii, where n = 5, is built up. First the 

 inner magic square of the (n — 2)th order is formed by any rule 

 we like to choose : the sum of the numbers in each line being 

 (n - 2) {(n — 2) 2 + 1}/2. To every number in it, 2n - 2 is added: 

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

 is now (n — 2) {n 2 +l}/2. The numbers not used are 1, 2,..., 



Figure xiii. A Bordered Square, n=5. 

 2n. — 2, and their complements, w 2 , n 2 — 1, . . ., n 2 — 2ra + 3. These 

 reserved numbers are placed in the 4 (n — 1) border cells so that 

 complementary numbers occur at the end of each row, column, 

 and diagonal of the inner square : this makes the sum of the 

 numbers in each of these latter lines equal to n (n 2 + 1)/2. It 

 only remains to make the sum of the numbers in each of the 

 border lines also have this value : such an arrangement is easily 

 made by trial and error. With a little patience a magic square 

 of any order can be thus built up, border upon border, and of 

 course it will have the property that, if each border is suc- 

 cessively stripped off, the remaining square will still be magic. 

 This is a method of construction much favoured by self-taught 

 mathematicians. 



Definite rules for arranging the numbers in the border cells 

 have been indicated*, though not, as far as I know, in a pre- 



* For instance, see Japanese Mathematics by D. E. Smith and Y. Mikami, 

 Chicago, 1914, pp. 116 — 120 : in this work, in the diagram on p. 120, the 

 numbers S and 29 should be interchanged. 



D. B. 



10 



