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I already know that 17 clues are a minimum for a Sudoku to have a solution, but I don't consider a Sudoku to be a real Sudoku while having more than one solution.

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    $\begingroup$ Wouldn't the minimum number of clues for a Sudoku to have a non-unique solution be 0? $\endgroup$ – noedne Apr 22 at 13:20
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    $\begingroup$ @noedne hm, so it did. I missed that when I looked at it. Good call. $\endgroup$ – Rubio Apr 22 at 19:07
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As long as there is a single Sudoku with 17 clues and a unique solution, the answer to your question is 17. And this paper has an example of such a puzzle, and its unique solution:

enter image description here

I was afraid this might end up needing some advanced trickery to complete, so I ran it through the online Sudoku 9x9.com solver. It made quick work of this, arriving at the (indeed) unique solution without having to use anything more exotic than naked and hidden singles and doubles.

See also this article wherein the author mentions collecting over 49,000 puzzles with 17 clues that are uniquely solvable, and discusses some of the commonalities found in them that guided the researchers who eventually proved that 17 clues is the minimum.

It should be noted that the very finding that "17 clues are a minimum for a Sudoku to have a solution" started from the premise that "the minimum number of clues needed to complete the puzzle, with only one unique solution, is 17, and puzzles with 16 clues or fewer must have multiple solutions." So the answer is actually inherent in your question.

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