Mathematical_Recreations-Kraitchik-2e

CHAPTER SEVEN
MAGIC SQUARES
1. DEFINITIONS
A MAGIC square is a square array of n2 distinct positive integers
with the property that the sum of the n numbers lying
in any horizontal, vertical, or main diagonal line is always the
same. In the accompanying illustration, for example, the
1 12 7 14
8 13 2 11
10 3 16 5
15 6 9 4
FIGURE 21.
horizontals give the sums 1 + 12 + 7 + 14 = 8 + 13 + 2 +
11 = 10 + 3 + 16 + 5 = 15 + 6 + 9 + 4 = 34, the verticals
give 1 + 8 + 10 + 15 = 12 + 13 + 3 + 6 = 7 + 2 + 16 + 9 =
14 + 11 + 5 + 4 = 34, and the main diagonals give 1 + 13 +
16+ 4 = 14+ 2+ 3+ 15 = 34.
It would be possible to admit negative numbers and fractions,
but no real generality would be gained thereby. On
the one hand, a magic square remains magic if the same number
is added to every number in the square, wherefore a
square with negative numbers can be changed to one with
only positive numbers by adding a sufficiently large positive
number to every number in the square. On the other hand,
a magic square also remains magic when everyone of its
numbers is multiplied by the same constant (not zero), wherelU