I don't know how magic squares would help. The constraints are totally different.

If you take a standard (N x N) magic square, consisting of the numbers from 1 .. N^2, and then reduce each number module N, you will in some (all?) cases get a square consisting of the numbers 1 .. N such that each number appears exactly once in each row and column.

The nature of your squares is identical to this case, except for the additional constraint that the N interior squares must each contain each number once. So solutions to this problem are a subset of the (n^2 x n^2) magic squares.

For this sort of problem I'm more used to either a) showing that there is a solution (any solution) by finding one; or b) finding all solutions. In your case you seem to be aiming at a quite different goal, in which randomness plays a necessary part, but I'm not at all clear what your precise goal is.

Finally, my concern at this point is not speed. Rather, it is the effectiveness of an algorithm to generate solutions of any size.

Hmm, so when failing to find a solution in 500,000 random attempts you're concerned not with how long that took but with how many attempts it took? But given that you're selecting the numbers at random, I'm not sure how you'd reduce that except by weighting the numbers towards the ones more likely to give a solution, and I can't offhand see any way to do that without knowing in advance what solutions are possible.

Sorry, I've realised that I really have no idea what you're asking for.

Hugo