matrices - Determinant of a 3x3 Magic Square

I don't have an explanation, but here is an outline of a proof (I've checked all the details myself, but it's laborious to write up correctly) that what you claim to happen actually does happen.

Result: Let $M$ be a 3x3 integer matrix whose columns, rows, diagonal, and anti-diagonal each total 15. The following are equivalent:

1. The entries of $M$ are distinct and positive;


2. The determinant of $M$ is $pm 360$;


3. Calling $h,i$ the (3,2) and (3,3) entries, $$(h,i)in {(1,6), (1,8), (3,4), (3,8), (7,2), (7,6), (9,2), (9,4)}.$$

The proof is just writing down the equations for the sums, rows, etc., to be 15 and solving, and then writing down the formula for the determinant and simplifying (it turns out $det = 45(h-5)(h+2i-15)$) and solving the resulting diophantine equation.

The explanation may well be just that there aren't very many 3x3 magic squares (basically, it looks like there's just one that we rotate and flip to get eight). For 4x4, the solution space will be 7d instead of 2d, and that is a lot of extra freedom. If you find a simply-described structure in the determinants of those, I'd be surprised.