Mathematical_Recreations-Kraitchik-2e

that is,
Magic Squares
b+p-Iu+p-1w == p+ 1
2 2 2 '
t+ v+ 1
a == 2 '
b==u+W+l,
2
modulo p,
modulo p.
161
Shifting the origin of the lattice will also ensure success
when both sets of diagonals are rows and columns of the fundamental
square. Figure 37 shows how the square of order 5
generated by ex, y) =
(1, 1) + rei, 2) r s(I, -2) can be made
magic by shifting rows and columns. The same effect would
have been produced by taking (4, 3) as the origin of the lattice.
Here the generators of the diagonals are (2, 0) and
(0,4).
15 21 7 18 4
24 10 16 2 13
8 19 5 11 22
17 3 14 25 6
1 12 23 9 20 1 12
15 21 7 18 4 15 21
24 10 16 2 13 24 10
8 19 5 11 22 8 19
17 3 14 25 6 17 3
FIGURE 37.
It is clear that distinct lattices may generate equivalent
squares. For example, the squares generated by
ex, y) = (a, b) + ret, u) + s(v, w)
and (x, y) = (b, a) + r(u, t) + sew, v)
will be equivalent. The curious reader might determine for
himself what effect the transformations generating equivalent
squares have on the generating lattice. By such means one