Mathematical_Recreations-Kraitchik-2e

166 Mathematical Recreations
sect in the cells occupied by the nine numbers, 68, 14,41,65,
11, 38, 71, 17, 44. The square will be magic if and only if one
of these numbers is at the center.
If a set of diagonals is generated by a motion (T, U), the
three numbers m, T, U cannot have a common divisor other
than 1. For it is easily shown that if they have, then the determinant
of the generating lattice, tw - uv, is not prime to
m. But it may happen that each of the numbers T, U has a
1 34 33 32 9 2
29 11 18 20 25 8
1 8 10 15 30 22 23 13 16 7
12 13 3 6 6 17 12 26 19 31
7 2 16 9 10 24 21 15 14 27
14 11 5 4 35 3 4 5 28 36
FIGURE 42.
different factor in common with m. For example, take the
semimagic square of order 15 generated by
(x, y) = (1, I)+r(4, 1)+ 8(-1,4),
whose diagonals are generated by (3, 5) and (5, -3) (Figure
41). The main upward diagonal, generated by
(x, y) = (1, 1) + r(3, 5),
forms in the fundamental square a rectangular lattice lying
at the points of intersection of columns 1, 4, 7, 10, 13, and
rows 1, 6, 11. All the other diagonals, of both kinds, follow
a similar pattern. In this particular case only those diagonals
which contain the number in the central cell of the
fundamental square, 113, are magic series, so the square can
be made magic in just one way.