Mathematical_Recreations-Kraitchik-2e

Magic Squares 187
From this one can readily verify that every magic square
contains 4 magic series in addition to the 10 prescribed. In
the notation of Figure 61 these are:
a+d+m+q=f+g+j+k
= b + c + n + p = e + i + h + l = s.
There are many other relationships connecting the elements
of a magic square of order 4. The following relations
are typical:
a + d = n + p, e + h = j + k, a + q = g + j,
a+b+e+f=c+d+g+h
i+j+ m+n= k+ l+ p+ q.
a b c S - (a + b + c)
e S - (a + b +e) a-c+e b+c-e
S S S
2"-c (a + b + c) - - 2" -a §. - b
2 2
S S S S
- - (a - c + e) - -(b + c - e) 2" - e (a + b + e) - 2"
2 2
FIGURE 63.
If we want the square to be panmagic, apparently six more
conditions are added, corresponding to the sums of the two
sets of three diagonals. Actually only three of these are independent,
so there are four independent constants left, and
the square may be written as in Figure 63.
5. ALGEBRAIC SQUARES. One of the above sets of relations
holding for every magic square of order 4 states that
the sum of the numbers in opposite quadrants of the square is
the same. If this sum is also the magic constant, we shall
call the square algebraic. This adds one further condition
on the constants of the square, so one of the constants disappears.
Such a square may be written as in Figure 64.