Mathematical_Recreations-Kraitchik-2e

Magic Squares 189
number of distinct none qui valent algebraic magic squares
is 3,456 + 8 = 432.
7. CLASSIFICATION OF NORMAL ALGEBRAIC SQUARES. The
distribution of the pairs of complementary numbers forms
the basis of a complete classification of algebraic squares.
Six types may be discerned, in the first three of which the
pairs of complements lie on diagonals, while in the others
they lie on orthogonals.
(1) Central. The complementary numbers are symmetrically
placed with respect to the center.
1 234
5 678
876 5
432 1
123 4
2 143
5 6 7 8
658 7
123 4
5 6 7 8
3 4 1 2
7 8 5 6
1 122
3 3 4 4
556 6
778 8
1 2 1 2
3 4 3 4
565 6
7 8 7 8
FIGURE 65.
122 1
3 4 4 3
5 6 6 5
788 7
(2) Diagonal. The complementary numbers form the diagonals
of the quadrants.
(3) Panmagic. A panmagic algebraic magic square is
characterized by the fact that the complementary numbers
lie on the diagonals in pairs whose elements separate each
other.
(4) and (5) Orthogonal (Adjacent and Alternating). The
complementary numbers lie in two columns, adjacent or
alternating.
(6) Symmetric. The complementary numbers are symmetric
to the vertical median.
If we designate a number and its complement by the same
digit, the six types may be represented as in Figure 65.