i
'

i G R S

il

i N S A

S
i
2
§.
i
é
i
i
&
|

 

154 THE FASCINATION OF NUMBERS

the total in sub-square G. Add the second number again to
the number just placed in G and place the new total in F.
This gives us:

 

in which the square contains three numbers. We now treat
each of these numbers in turn. Add the third number (g) to
the first number (47) and place the total in sub-square K.
Add g again and write the result in D. Add g to the number
in G and place the result in E. Again add g to the number
written in E and write the result in C. Now add g to the num-
ber in F and place the result in A. Add g again and place the
result in H. All sub-squares are now filled and we have:

oo |57

The reason why this must be a magic square is best demon-
strated by showing how each individual number is obtained.

Other squares have been constructed using only prime
numbers, and it is possible to construct what are called
doubly-magic squares, the peculiar property of which is that,
if the original integers are squared, the resulting square is
also magic.

Finally, we may refer to subtracting and multiplying
magic squares. In the former type of square, a constant
‘total’ results for each row, column, or diagonal by deducting