Just a guess. Perhaps you could use the other monk's blanked out solution to create an initial grid. Then, recreate this solution in the other grids, but transform it in some simple way in order to make sure it doesn't break the rules. My shot  just a first guess  is below, hidden:
Hmm. Let's try a 2 x 2 version. The initial grid is
12
34
obviously. Now you need to fill in
12
34


Clearly, as you move across, you need to shift rows:
1234
3412


And as you move down, you need to shift columns:
1234
3412
21
43
note that the lower left grid is just the upper left grid with columns shifted along.
Finally,
1234
3412
2143
4321
it turns out that the column shift from the grid above, and the row shift from the grid to the left, give identical results.

Perhaps this will work with N>2. Or not. Enjoy.
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