Mathematical_Recreations-Kraitchik-2e

168 M athematiral Recreations
one by reversing the procedure - starting with a given
magic square and bordering it.
Let us begin with a square of order 4 (Figure 42) using the
numbers 1 to 16. We must supply 6 2 - 42 = 20 numbers.
First increase each number in the original square by 20 + 2
= 10. Then we must try to insert the numbers 1 to 10 and
7 32 6 18 35 13
1 29 10 9 26 36
34 20 15 16 23 3
33 14 21 22 17 4
12 11 28 27 8 25
24 5 31 19 2 30
FIGURE 44.
7 36 8 11 31 18
9 2 33 34 5 28
27 17 22 21 14 10
25 23 16 15 20 12
24 32 3 4 35 13
19 1 29 26 6 30
FIGURE 45.
27 to 36 into the border. Arrange these numbers in complemen
tary pairs:
1
36
2
35
3
34
4
33
5
32
We take care of the diagonals by inserting two complementary
pairs at the diagonal corners. Then we select numbers
to fill in the top row and right-hand column so as to give the
magic constant, and set their complements orthogonally opposite
them.
By repeating this procedure we can form magic squares
of order 8, 10, 12, .... Figure 43 represents a magic square
of order 12 obtained from the square of Figure 42 in this way.
The squares we have formed consisted of consecutive numbers,
but that is not nece'Ssary. In Figures 44 and 45 we give
two border squares whose interior squares do not have consecutive
numbers.
6
31
7
30
8
29
9
28
10
27