Mathematical_Recreations-Kraitchik-2e

Magic Squares 145
change of rows may be expressed geometrically by supposing
the array to be inscribed on a cylinder so that the top and
bottom rows meet. Any row may then be considered the
bottom row. Cyclic interchange of both rows and columns
B A 0 C
~ ~
C 0 A B
FIGURE 24.
Order.
Transformation by Interchange of Quarters, Even
may be obtained by inscribing the array on a torus (anchor
ring).
A magic square remains magic if any two rows equidistant
from the center are interchanged, provided that the two colo
c
C
b
A
a
B
FIGURE 25.
Order.
Transformation by Interchange of Quarters, Odd
umns at that same distance from the center are also interchanged.
Figure 23 shows the manner in which such a
transformation affects the elements at the intersections of
the transposed rows and columns. Repetition of this transformation
for all such sets of four orthogonals results in a rotation
of the original square through 180°.