Mathematical_Recreations-Kraitchik-2e

158 Mathematical Recreations
mental series only if they lie in the central column, that is, if
b = P ; 1.
Similarly, if u alone is divisible by p the lattice
will cover the cells of a row of the fundamental square, and
the corresponding numbers will form a magic series only if
this is the middle row, for which a = p; 1.
Clearly, if
both t and u are divisible by p the lattice will cover only the
cell Ca, b) in the fundamental square, and will be quite useless.
The basic idea of the method of finding regular magic
20
9 '
s-
21 22 2'3 24 ~~
161~. 1819 ZI!
1/ 12 13rJ4 IS
tr 7 . .B' eg 10
.'J, "2 3 4 5
4
22
18
·11 , ,14
_ 7'
3
21 24
17·
~
.2
2S
16
10
FIGURE 34,
13
20 22 4 6 13
9 11 18 25 2
23 5 7 14 16
12 19 21 3 10
Magic Square and Lattice.
1 8 15 17 24
squares now begins to show itself. We want to cover the cells
of the fundamental square with a two-dimensional lattice,
(x, y) = Ca, b) + rCt, u) + s(v, w), whose rows, columns, and
principal diagonals are magic series. To make sure that the
lattice covers every cell of the fundamental square we must
select t, u, v, w so that tw - uv is not divisible by p; otherwise
we shall find that only p of the cells are covered. If this
has been taken care of, and if p does not divide t, u, v, or w,
the square whose rows and columns are the rows and columns
of the lattice will then be at least semimagic. Figure 34