Log24

Unmagic Squares

Consecutive positive integers:

1   2   3
4   5   6
7   8   9

Consecutive nonnegative integers:

0   1   2
3   4   5
6   7   8

Consecutive nonnegative integers
written in base 3:

00  01  02
10  11  12
20  21  22

This last square may be viewed as
coordinates, in the 3-element Galois
field GF(3), of the ninefold square.

Note that the ninefold square so viewed
embodies the 12 lines of the two-dimensional
affine space over GF(3)…

As does, similarly, the ancient Chinese
"magic" square known as the "Lo Shu."

These squares are therefore equivalent under
affine transformations.

This method generalizes.

— Steven H. Cullinane, Nov. 20, 2021