In the n-by-n case when n is even, all of them are possible :) In the n-by-n case when n is odd , that's not the case... it breaks into a few equivalence class

In the n-by-n case when n is even, all of them are possible :) In the n-by-n case when n is odd , that's not the case... it breaks into a few equivalence classes that you can separate out by looking at the "mod 2 sums" across each of the rows and columns.