Minimum number of givens for General Sudoku of size $n^2 \times n^2$

Question

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.

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. — D.W., Mar 13 '15 at 16:44
@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. — Ryan Dougherty, Mar 13 '15 at 16:45
Possible duplicate of Minimum number of clues to fully specify any sudoku? — David Richerby, Sep 30 '17 at 18:29
@DavidRicherby, that question asks about 9x9 sudoku, while this asks about generalization. — rus9384, Oct 06 '17 at 01:03

tomoka kazuki · Answer 1 · 2017-10-12T00:51:13.243

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.

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? — rus9384, Oct 07 '17 at 17:06
Seems like an open problem well into to the delve of research. — User Not Found, Dec 31 '17 at 13:47

Minimum number of givens for General Sudoku of size $n^2 \times n^2$

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