Mathematical_Recreations-Kraitchik-2e

180 Mathematical Recreations
onals we need only require that tw - UV, t, u, v, wall be prime
to n. If we make the further requirement that the coefficients
of the motions generating the diagonals also be prime
to m, we obtain Euler squares that are not only diagonal, but
pandiagonal. These can be obtained for every odd order n
except those that are divisible by 3.
The fact that" regular" Euler squares can be formed subject
to fewer restrictions than are applied to the formation
of regular magic squares of the corresponding orders might
nl n2 n3 n4 n5 nn
31 32 33 34 35 3n
21 22 23 24 25 2n
11 12 13 14 15 In
FIGURE 56.
lead one to suppose that there are more Euler squares than
magic squares. The contrary is actually the case. Every
Euler square gives rise to a semimagic square, but many irregular
magic squares cannot be translated into Euler squares.
Except for the condition that the elements of a magic
square be distinct numbers, the requirements on the elements
in a magic square are purely quantitative. The conditions
on the elements of an Euler square, on the other hand, may
be expressed purely qualitatively. As in the case of the problem
of the 36 officers, the objects to be arranged may be any
objects at all that are capable of being described in terms of
two qualities, each of which has n categories. The objects