## The Determinants of 4x4 Magic Squares

Up to sign, there are only 12 distinct determinants for 4x4 magic
squares (using the elements 0 to 15). The numbers of squares having
each of these determinants are listed below:
exponents in number of number of
factorization squares squares
|det| 2 3 5 7 |det| -|det|
------- ------------- -------- --------
0 - 5120
1920 7 1 1 0 160 160
2880 6 2 1 0 64 64
3840 8 1 1 0 96 96
4800 6 1 2 0 32 32
5760 7 2 1 0 192 192
6720 6 1 1 1 32 32
7680 9 1 1 0 64 64
8640 6 3 1 0 32 32
9600 7 1 2 0 64 64
11520 8 2 1 0 96 96
15360 10 1 1 0 128 128
In a sense we can say the 4x4 magic squares are split into 12 distinct
sets (or 23 sets counting differences in sign). In each set, the
number of squares is one of these values:
32 = (2^5) = 1(32)
64 = (2^6) = 2(32)
96 = (2^5) (3) = 3(32)
128 = (2^7) = 4(32)
160 = (2^5) (5) = 5(32)
192 = (2^6) (3) = 6(32)
5120 = (2^10)(5) = 160(32)
The squares with negative determinant are evidently related
to those with positive determinant by taking the SUM complement
of each element.
I wonder if all the magic squares (of a given order) with the
same determinant constitute a group under some set of "simple"
transformations. It's certainly true for the sets of 32, since
these can be shown to constitute a group under rotation, reflection,
etc. But what about the larger sets?
Interestingly, the unique (up to sign and 32-group transformations)
square with determinant 6720 is the only square whose determinant
is divisible by any prime other than 2, 3, or 5.
It's also interesting to experiment with various matrix operations
on two magic squares A,B. For example, the squares given by ABA^-1
have the same row and column sums as A and B. The commutator
AB-BA also gives some interesting results, as does replacing each
element of a square A with its co-factor.

Return to MathPages Main Menu