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It is well known (as shown here) that the minimum number of starting clues a Sudoku puzzle may have to generate a unique solution is 17.

My main question is

Given a completed Sudoku grid, is it always possible to find a subset of 17 starting clues which generate the grid uniquely?

I suspect the answer might be 'no' as, accounting for symmetries, there are 3,359,232 distinct Sudoku grids and I think there are only around 50,000 known 17-clue Sudoku puzzles. This leads me to a follow-up question.

Let $\mathbb{S}$ be the set of all completed Sudoku grids. For a given $S \in \mathbb{S}$, let $m(S)$ be the minimum number of starting clues required to generate $S$ uniquely.
What is $$M = \max\{ m(S) | S \in \mathbb{S}\}?$$

If the answer is not known, do we have an upper bound for $M$?


What do I know so far?

We can demonstrate, by hand, that $M \leq 60$.
Take any completed Sudoku grid and remove the following entries (marked by an X):

enter image description here

The solution to this Sudoku is fully determined and some subset of the remaining clues will constitute a minimal set of starting clues.


I am almost certain that $M < 40$.

According to the mathematics of Sudoku, "The most clues for a minimal Sudoku is believed to be 40, of which only two are known." One way of generating Sudoku puzzles would be to start with a completed grid and remove entries using some algorithmic procedure until a minimal set of clues was reached. So much analysis has been done on this that, in my opinion, if somebody encountered a grid where they could not generate a set of starting clues with less than 40 entries, this would be of considerable note.


I suspect that $M$ is around $20$ and it's conceivable to me that somebody may have indirectly encountered a much better upper bound for $M$ while trying to generate puzzles from completed grids.


Update: A proof that $M \leq 48$ was given here on Puzzling SE

Also, $M \geq 18$, as verified by examples given here

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  • $\begingroup$ There's a trivial upper bound $M \le 64$. $\endgroup$ Commented Jun 26, 2019 at 8:51
  • $\begingroup$ @PeterTaylor My thinking is that we may be able to say $M \leq 40$ as the most clues for a minimal Sudoku is believed to be $40$ although I don't know if this has been proven. $\endgroup$
    – hexomino
    Commented Jun 26, 2019 at 9:05
  • $\begingroup$ @PeterTaylor In fact, I can show quite easily that $M \leq 60$. Do you think it's worth adding this to the question? $\endgroup$
    – hexomino
    Commented Jun 26, 2019 at 11:54
  • $\begingroup$ Definitely. The context of what you already know about the question nearly always improves it. $\endgroup$ Commented Jun 26, 2019 at 12:50
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    $\begingroup$ @Sigur On the other extreme, it's possible to have a Sudoku grid with 77 clues and multiple possible solutions. Number of clues says very little if those clues aren't optimally distributed. $\endgroup$
    – Brilliand
    Commented May 19, 2020 at 17:52

1 Answer 1

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I guess the page https://en.wikipedia.org/wiki/Mathematics_of_Sudoku says ,,A puzzle with a unique solution must have at least 17 clues, and there is a solvable puzzle with at most 21 clues for every solved grid.'' As I understand the second part of the sentence means $M \le 21$ in this context.

Don't know much about it, but didn't see it mentioned in the answers. (I myself was googling if the 40 bound for the minimal sudoku was already proved or disproved)

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  • $\begingroup$ Oh, yes, I think you're right. This wasn't on the page when I posted my question. Do you know when it was updated? The source seems to have originated here - the day after I posted this question! $\endgroup$
    – hexomino
    Commented Feb 25, 2022 at 13:07
  • $\begingroup$ I think the post here seems to verify that M=21. Is that right? $\endgroup$
    – hexomino
    Commented Feb 25, 2022 at 13:10

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