Is there an algorithm to generate a sudoku puzzle having the minimum number of entries that has exactly $k>1$ solutions? What is this minimum number of entries, as a function of $k$? Given that the minimum number of entries for $k=1$ solutions is $17$, then surely the solution to this question is less than or equal to $17$.
This question has been raised in 2006 on this forum. Even "the big guys" of sudoku maths (Ed Russell and Fredrik Kjell) were unable to give a real answer.