Inspired by this question,
consider hints on a Sudoku board. A regular puzzle has a unique solution.
It is clear that there are puzzles with 2 or 3 solutions, and therefore, I guess, puzzles with say 4, and 6 solutions.
Now, what is the smallest integer k such that there is no set of Sudoku clues resulting in exactly k solutions?
Like it usually happens with these conjectures, if the solution is not k=7 or k=11, it may be really hard to find. Of course, there exists a solution, for there is a maximum (huge) number of solutions that a blank sudoku allows. A property that may help to construct solutions is that for small values, if we have a sudoku that allows a solutions and another that allows b solutions, we can probably find a sudoku that allows a·b solutions. For example:
This allows k=4 fillings:
And this allows k=3 fillings:
From this, we can easily see that this:
allows k = 4·3 = 12 fillings.
Although it is hard to generalize, probably the number you are looking for is prime, because if it was to be a composite number, its factors (smaller than the number) would have to be a number of fillings for some sudoku, and then this non-rigorous multiplication rule would fail.