Mathematical_Recreations-Kraitchik-2e

Magic Squares 163
semimagic if t, u, v, ware all prime to m. The square will be
magic if the origin is chosen so that the central cell of the
fundamental square becomes the central cell of the new
square, and it will be panmagic if t + v, t -
v, u + w, and
u - ware all prime to m.
Thus we can have a panmagic square of order 35 generated
by (x, y) = (1, 1) + r(2, 1) + s(I, 2). The square is panmagic
because its diagonals are generated by (3, 3) and (1, -1).
If the square is not panmagic the situation may be more
complicated than when the order is prime. Whereas pre-
73 74 75 76 77 78 79 80 81
64 65 66 67 68 69 70 71 72
55 56 57 58 59 60 61 62 63
46 47 48 49 50 51 52 53 54
37 38 39 40 41 42 43 44 45
28 29 30 31 32 33 34 35 36
19 20 21 22 23 24 25 26 27
10 11 12 13 14 15 16 17 18
1 2 3 4 5 678 9
81 10 29 48 67 5 24 43 62
71 9 19 38 57 76 14 33 52
61 80 18 28 47 66 4 23 42
51 70 8 27 37 56 75 13 32
41 60 79 17 36 46 65 3 22
31 50 69 7 26 45 55 74 12
21 40 59 78 16 35 54 64 2
11 30 49 68 6 25 44 63 73
1 20 39 58 77 15 34 53 72
FIGURE 39.
viously the diagonals that caused the trouble were always
rows or columns in the fundamental-square, this is no longer
the case. Suppose a set of diagonals is generated by (T, U).
If one of the direction numbers is divisible by m the corresponding
diagonals will be rows or columns of the fundamental
square, as before. But if one of the direction numbers,
though not divisible by m, has a factor in common with
m, then the corresponding diagonals of the new square are
neither rows nor columns of the fundamental square, but they
form in it a set of one- or two-dimensional lattices with sides
parallel to the axes. For example, if m = 9 and we generate
a semimagic square by the lattice (x, y) = (1, 1) + rei, 2)+
s(l, 1), the set of upward diagonals is generated by (2, 3),