Mathematical_Recreations-Kraitchik-2e

Magic Squares 169
One can also border squares of odd order. Figure 46 shows
a square of order 9 obtained by successively bordering a
square of order 3.
According to Frolow all possible ways of bordering the
normal magic square of order 3 are obtained from the 10
borders given in Figure 47 by (a) permuting the middle three
rows, (b) permuting the middle three columns, and (c) sub-
16 81 79 78 77 13 12 11 2
76 28 65 62 61 26 27 18 6
75 23 36 53 51 35 30 59 7
74 24 50 40 45 38 32 58 8
9 25 33 39 41 43 49 57 73
10 60 34 44 37 42 48 22 72
14 63 52 29 31 47 46 19 68
15 64 17 20 21 56 55 54 67
80 1 3 4 5 69 70 71 66
FIGURE 46.
jecting each border to the 8 transformations described on
page 144. Since there is just one square of order 3, we get
10·(31)2·8 = 2,880
nonequivalent normal border squares of order 5.
These are not all, however, since these include only those
border squares of order 5 formed from a magic square of
order 3 composed of the numbers 9 to 17, the middle nine of
the first 25 numbers. Violle has completed the study of the
question by showing that there are 26 different magic squares
that can be formed with 9 of the first 25 numbers, and by