Mathematical_Recreations-Kraitchik-2e

186 Mathematical Recreations
contains two even numbers and two odd, except those arithmetic
series marked with (e) and (0) respectively.
Except for the symmetric series, magic series occur in complementary
pairs, the numbers of one series of such a pair
being the complements of the numbers of the other series.
a b c d
e 1 g h
i j k l
m n p q
FIGURE 61.
4. LITERAL MAGIC SQUARES. Let US determine what relations
the elements of every magic square of order 4 (not
necessarily normal) must satisfy. For this purpose we shall
a b c 8 -(a + b + c)
e 1 g 8 -(e +1 + g)
i
2a+b+c+e 28 - 2a - b -
-g+i-8 c-e-I-i
28 - 2a - 2b
8 -(a + e + i) -c-e-I+
g - i
FIGURE 62.
I+g-i
2a + b + e +1 a+b+c+e
-g+i-8 +i - 8
write the square with letters, as in Figure 61. By definition
the values of 10 sums (the sums of the 4 rows, of the 4 columns
and of the 2 diagonals) are fixed. But the sum of the
sums of the rows is identical with the sum of the sums of the
columns, so that only 9 of the 10 relations are independent.
There are thus 16 - \} = 7 independent elements (and the
magic constant, S). We find that the square may be written
as in Figure 62.