Mathematical_Recreations-Kraitchik-2e

Magic Squares 191
We may remark in passing that the three kinds of symmetric
squares are very similar, and Frenicle put them into
a single class. There are 304 of them in all.
Central:
1 8 12 13 1 8 14 11 1 8 15 10
14 11 7 2 12 13 7 2 12 13 6 3
15 10 6 3 15 10 4 5 14 11 4 5
4 5 9 16 6 3 9 16 7 2 9 16
Panmagic:
1 8 10 15 1 8 11 14 1 8 13 12
12 13 3 6 12 13 2 7 14 11 2 7
7 2 16 9 6 3 16 9 4 5 16 9
14 11 5 4 15 10 5 4 15 10 3 6
1 4 16 13 1 6 16 11 1 7 16 10
Adjacent 14 15 3 2 12 15 5 2 12 14 5 3
Orthogonal: 7 6 10 11 7 4 10 13 6 411 13
12 9 5 8 14 9 3 8 15 9 2 8
Symmetric:
1 4 13 16 1 6 11 16 1 7 10 16
14 15 2 3 12 15 2 5 12 14 3 5
8 5 12 9 8 3 14 9 8 2 15 9
1110 7 6 13 10 7 4 13 11 6 4
FIGURE 66.
9. If the magic squares are not normal there are three
cases to consider.
First, if the given 16 numbers can be arranged so as to form
an addition table whose components satisfy the relations
A + D = B+ C, a+ d = b+ c, then the 528 algebraic and
derivative symmetric squares can be formed, but in general
no arithmetic squares exist.