Mathematical_Recreations-Kraitchik-2e

164 Mathematical Recreations
and the set of downward diagonals is generated by (0, 1).
(The fundamental square and the corresponding semimagic
square generated by this lattice are shown in Figure 39.)
It will be seen that the downward diagonals, generated by
(0, 1), are columns of the fundamental square. As in the
case of prime order, only the diagonal corresponding to the
central column of the fundamental square will be a magic
68 76 3 11 19 36 44 52 60
54 62 70 78 5 13 21 29 37
31 39
47
55
72 80 7 15 23
31 39
17 25
33
41
49 57 65 73 9
17 25
75 2
10
27
35 43 51 59 67
75 2
61 69
77
4
12 20 28 45 53
61 69
38 46
63
71
79 6 14 22 30
38 46
24 32
40
48
56 64 81 8 16
24 32
1 18
26
34
42 50 58 66 74
1 18
3
11
19 36 44 52 60
68 76
70
78
5 13 21 29 37
54 62
FIGURE 40.
series. But the upward diagonals, generated by (2, 3), are
very different in appearance. For example, the diagonal
generated by (x, y) = (3, 2) + r(2, 3) is composed of three
one-dimensional lattices in three of the rows of the fundamental
square: 12,15,18 in the second row; 38,41,44 in the
fifth row; and 64,67,70 in the eighth row.
If we form the sums of the upward diagonals we find that
there are three magic series among them, and that these intersect
the only downward magic diagonal at the numbers 68,