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Here, it is well known that the minimal number of givens for a size $9 \times 9$ board of Sudoku requires 17 "givens" in order to be solved (i.e., no puzzle can be solved with $\le 16$ givens).

What is the minimal number of givens for a size $n^2 \times n^2$ board? Is there a table of "best-known" minimal values?

I'm speculating that there is only an asymptotic/approximate bound as the $n=3$ case was not known, let alone higher values of $n$. For $n=4$, the best-known is 55, and $n=5$ is 151.

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    $\begingroup$ That link says the proof that 17 "givens" are required was only obtained through exhaustive computer search. Therefore, there's not likely to be a known formula that works for general $n$; and when $n$ is large, exhaustive search to find the exact answer is not likely to be feasible. Asking for a table of best known bounds is reasonable, though. $\endgroup$
    – D.W. ♦
    Mar 13, 2015 at 16:44
  • $\begingroup$ @D.W. I was just wondering if there is a known proof that at least $f(n)$ givens is required or something. A table would be more helpful though. $\endgroup$
    – Ryan
    Mar 13, 2015 at 16:45
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    $\begingroup$ Possible duplicate of Minimum number of clues to fully specify any sudoku? $\endgroup$
    – David Richerby
    Sep 30, 2017 at 18:29
  • $\begingroup$ @DavidRicherby, that question asks about 9x9 sudoku, while this asks about generalization. $\endgroup$
    – rus9384
    Oct 6, 2017 at 1:03
  • $\begingroup$ oeis.org/A198297 $\endgroup$
    – Pseudonym
    Oct 8, 2017 at 3:09

1 Answer 1

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The minimum number of clues for a puzzle of $n^2$ x $n^2$ is known for $n = 2$ (4 x 4 Sudoku), of which the minimum clues is 4 (25% of cells filled). There are 13 such non-equivalent puzzles. (based on info at "enjoysudoku.com").

The minimum number of clues for a puzzle of $n^2$ x $n^2$ is also known for $n = 3$ (9 x 9 Sudoku), of which the minimum clues is 17 (about 21.0% of cells filled). About 49,000 such non-equivalent Sudokus have been discovered.

For $n = 4$ (16 x 16), the fewest clues known in any puzzle is 55 (about 21.5% of cells filled), but it is not known if this is the fewest possible. (based on info at "enjoysudoku.com").

For $n = 5$ (25 x 25), little work has also been done to find the fewest clues possible, and comparatively few Sudokus of this size have been constructed. One 25 x 25 Sudoku is known to have 328 clues. So from a cursory study, the fewest clues for (n=5) is 328 or fewer (about 52% or fewer cells filled). Note: this Sudoku was not an attempt to be an example with few clues. I listed it only to provide an initial upper bound.

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  • $\begingroup$ For n in {0,1}, the minimum is zero. ​ ​ $\endgroup$
    – user12859
    Oct 6, 2017 at 2:14
  • $\begingroup$ Well, is it proven at least that number of givens tends to become closer to number of cells as $n$ grows? Maybe some upper bound on limit ($\le 1$) is known? Or $1$ is tight bound? $\endgroup$
    – rus9384
    Oct 7, 2017 at 17:06
  • $\begingroup$ Seems like an open problem well into to the delve of research. $\endgroup$
    – User Not Found
    Dec 31, 2017 at 13:47

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