33 845 721 combinations of pieces around top-left clue piece (4x4 puzzle)
33 845 721 combinations of pieces around upper-left clue piece (4x4 puzzle), with the 4 clue pieces and the mandatory piece fixed.
If only mandatory piece was fixed (and the top-left clue piece of course), this would give 37 917 149 combinations.
With the mandatory piece unfixed (yes, this is stupid), this 4x4 partial puzzle would have 39 298 666 combinations.
In each case, using a 4x4 partial puzzle reduces considerably the number of combinations around the clue piece compared to the 3x3 partial puzzle.
If only mandatory piece was fixed (and the top-left clue piece of course), this would give 37 917 149 combinations.
With the mandatory piece unfixed (yes, this is stupid), this 4x4 partial puzzle would have 39 298 666 combinations.
In each case, using a 4x4 partial puzzle reduces considerably the number of combinations around the clue piece compared to the 3x3 partial puzzle.