Mathematical_Recreations-Kraitchik-2e

Magic Squares 177
square of order 128. His method was improved by Cazalas,
who formed trimagic squares of orders 64 and 81. Mr. Royal
V. Heath has formed many bimagic squares, and a trimagic
square of order 64 different from Cazalas' square.
Here is a very simple method (due to Aubry) for the formation
of a bimagic square of order 8.
Let a, b, c, d represent the numbers 0, 3, 5, 6 in some order,
and let A, B, C, D be numbers determined by the relations
A + a = B + b = C + c = D + d = 7. These eight numbers
form a multigrade of second order (see p. 79), since
a+b+c+d=A+B+C+D
and a2 + b 2 + c 2 + d 2 = A 2 + B2 + C2 + D2.
Symbolic expressions such as the following will be called
bimagic lines of 8:
I
II
III
IV
Oa Ib 2c
Oa Ib 2c
Oa Ie 2A
Oa IA 2b
3d 4d
3d 4A
3C 4b
3B 4c
5e 6b 7a
5B 6C 7D
5d 6B 7D
5C 6d 7D
If we interpret each term (Oa, 1b, 2c, and so on) as a product,
then the sum of the terms in each line is the same. If we
interpret each term as a two-digit number in the scale of notation
with base 8, then their sum, and the sum of their
squares, is constant. These results follow from the given
relations connecting a, b, ... , D, and are true no matter how
the values of a, b, c, d are chosen from among the numbers
0, 3, 5, 6.
Other magic lines of 8 can be formed from these by permuting
the letters among themselves and the digits among
themselves. For example, if we interchange the numbers of
the pairs (0, 1), (2,3), (4,5), (6,7) in II we get la, Ob, 3c, 2d,
5A, 4B, 7C, 6D. Such a permutation will be denoted: (01),
(23), (45), (67). We shall use the following permutations of
this sort: