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I've seen that the fewest clues on a Sudoku board has been proven to be 17 but I'm wondering if it's possible for every board to have some combination of 17 clues or, if not, if there is a proven minimum for every possible board.

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    $\begingroup$ Weirdly, I've asked the same question before on Math SE: math.stackexchange.com/questions/3274039/… although I didn't get much of a response. Given that the number of known 17-clue Sudokus is much less than the number of possible Sudokus, it looks like 17 is not always possible but what the minimum is, I have no idea. $\endgroup$
    – hexomino
    Aug 13, 2021 at 9:17
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    $\begingroup$ Related: Sudoku net that is always solvable $\endgroup$
    – bobble
    Aug 13, 2021 at 13:59

1 Answer 1

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Try
123456789
456789123
789123456
231564897
564897231
897231564
312645978
645978312
978312645

167238459
258149367
349567128
716823945
825914736
934756812
671392594
582481673
493675281

If I didn't make any mistakes constructing them, they can't have unique solutions with 17 or fewer clues. I don't know if there's an 18 clue puzzle for them offhand.

For the first, look at the 1, 2, and 3 in R147 C123. It takes at least 2 clues to be able to get all of those. (If you only had R1C1 and every digit not in those nine, the solution wouldn't be unique.) The whole grid can be divided into nine sets that each require at least two members must be given as clues. That requires 18 clues.

In the second one, there are sets of four requiring one clue (R12 C14), sets of six requiring one clue (R3C78, R6C89, R9C79), sets of nine requiring 2 clues (R123 C369).

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