Mathematical_Recreations-Kraitchik-2e

Magic Squares 159
shows how this method is applied when p = 5, using the lattice
(1, 1) + r(2, 1) + s(l, 2).
Of course it is not sufficient to have magic series in the rows
and columns. We must also take care of the diagonals. The
set of upward diagonals is generated by the motion (t + v,
u+ w), and the other set by (t - v, u - w). If it happens
that none of the numbers t + v, t - v, u+ w, u- w is divisible
by p, then every diagonal in each direction will be a magic
series, so the square will be not only magic, but panmagic.
That is the case with the square in Figure 34 whose diagonals
43 44 45 46 47 48 49
36 37 38 39 40 41 42
29 30 31 32 33 34 35
22 23 24 25 26 27 28
15 16 17 18 19 20 21
8 9 10 11 12 13 14
123 4 5 6 7
20 28 29 37 45 4 12
32 40 48 7 8 16 24
44 3 11 19 27 35 36
14 15 23 31 39 47 6
26 34 42 43 2 10 18
38 46 5 13 21 22 30
1 9 17 25 33 41 49
FIGURE 35.
are generated by (3, 3) and (-1, 1). It is also the case in the
square of order 7 (Figure 35) generated by the lattice (x, y) =
(1, 1) + r(l, 1) + s(2, -2). Its diagonals are generated by
(3, -1) and (-1,3).
Suppose, however, that one set of diagonals is composed
of rows or columns of the fundamental square. For example,
in the square of order 5 determined by ex, y) = (1, 1) +
reI, 2) + s(l, -1), the diagonals are generated by (2, 1) and
(0, 3). (This square is shown in the upper 5 rows of Figure
36.) Here all the upward diagonals are magic series, but
only one of the downward diagonals is, and that one is not a
main diagonal. The one downward diagonal which is a magic
series is, we know, the one that consists of the central row or