Mathematical_Recreations-Kraitchik-2e

170 Mathematical Recreations
finding for each case the total number of possible borders.
He finds 594·288 = 171 ,072 squares in all,
8. COMPOSITE MAGIC SQUARES
A composite m X n magic square is a magic square of order
mn which can be decomposed into m2 magic squares, each of
3 22 21 18 1 3 22 20 19 1 6 24 23 8 4 62523 7 4
7 19 8 18 25 1 24 2
24 2 24 2 7 19 5 21
6 20 5 21 5 21 8 18
25 4 5 8 23 25 4 6 7 23 22 2 3 18 20 22 1 3 19 20
72524 6 3 7 22 25 8 3 7 6 23 24 5 7 25 24. 4 5
22 4 6 20 25 23 3
8 18 5 21 8 18 8 18
5 21 24 2 4 22 6 20
23 1 2 20 19 23 4 1 18 19 21 20 3 2 19 21 1 2 22 10
82523 7 2 824 23 4 6
22 4 25
5 21 5 21
6 20 7 19
24 1 3 19 18 20 2 322 18
FIGURE 47.
order n. A general method of -constructing such squares is
shown in the following examples.
We begin with the normal-magic square of order 3 (Figure
48). In this square we shall replace each element by a square
of order 3 as follows: Replace 1 by the square itself. Replace
2 by the square obtained by adding 9 to each element
of the given square. Replace 3 by the square obtained by
adding 18 to each element of the given square. In general,
replace each number k in the given square by the square obtained
from the given square by adding (k - l)n2 to each