Mathematical_Recreations-Kraitchik-2e

144 Mathematical Recreations
replacing each of its numbers by its second power is also
magic., A bimagic square is trimagic if the third powers of
its elements form a magic square. And so on. Such squares
may be called multimagic.
2. TRANSFORMATIONS
Any square array may be subjected to certain geometric
transformations which affect one's manner of looking at the
FIGURE 22.
Reflection.
Transformations of a Square by Plane Rotation and
array rather than the interrelations of the objects contained
in it. These transformations form a group generated by two
fundamental operations - rotation through a right angle
x y y x
y Q b x c d
x d c y b a
FIGURE 23. Transformation by Interchange of Rows.
and reflection in a mirror. Starting from anyone array,
seven others may be obtained in this manner. Figure 22
shows such a set. Any two members of such a set may be
called equivalent. In discussing magic squares equivalent
squares are usually not considered as distinct.
Sometimes a square array keeps a distinctive property (for
example, a panmagic square remains panmagic) under cyclic
interchange of rows or of columns or both. Cyclic inter-