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The answer is $$6\, 670\, 903\, 752\, 021\, 072\, 936\,960$$ according to this site:

https://www.technologyreview.com/2012/01/06/188520/mathematicians-solve-minimum-sudoku-problem/

I have tried to get this number using direct methods but basically I have found the question too hard. There are too many possibilities, but I am likely missing some strategic ways to solve the problem. Any help would be greatly appreciated, thanks.

calculatormathematical
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Adam Rubinson
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4 Answers4

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The Link has change to: Enumerating Possible Sudoku Grids. Although enumerating implies listing them all, which they didn't.

The second write up they changed the name to Math of Sudoku I.

The question is vague enough to note that Nmax is 6, 670, 903, 752, 021, 072, 936, 960 and Nmin is the number of unique equivalence classes which is 5,472,730,538. Multiply this by 3,359,232 reorderings and 9! relabelings and you get roughly Nmax plus some extra caused by over-counting automorphisms.

But the question is actually how may sudoku puzzles, which none of these answers... answers.

You can start by taking all girds, Nmax and subtracting 1 clue from each getting 81xNmax. Then you can start again and take subtract 2 clues so there is C(81,2)xNmax. Then C(81,3)xNmax, etc. and sum them all. So Sigma of N=17 to 81 of C(81,n) will get you close, but you will need a Big Int package to compute this.

You could substitute Nmin here to save some work. ;')

But there is the question of solvability. If there are 4 or more missing clues you have to deal with unavoidable loops, where one of the four has to be filled for the puzzle to be solvable. And there are loops greater than four.

Then somewhere along the way you start having unsolvable puzzles with a large number of clues, sort of a max-min situation. Maybe for reasons other than unavoidable loops?

Then down at the low end you need at least 17 clues remaining to have a solvable puzzle. There are 49158 17 clue puzzles and they have all been found. Work is ongoing to count 18 clue puzzles, which don't contain 17 clue puzzles.

So it might be a while before we have an actual answer to this question.

Sojourner9
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0

Try Wikipedia for an introduction and overview and additional references.

Hagen von Eitzen
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  • Somehow I'm missing the explanation for the bug number there. I had thought about it myself, and thought like this: For the first row there are 9! possibilities, and for the second row there are 8! possibilities (each column has one possibility less). Then by induction I would get "9! * 8! * 7! 6! 5! 4! 3! 2! 1!", or near 1.83 * 10^21, but the official number is higher. – U. Windl Oct 15 '23 at 22:49
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I think it would be (9!)squared X 9 The 9 is from the 9 different puzzles with the center number changed. Then you would have to subtract all the puzzles with 0 thru 7 solved numbers(if you had the 8th solved, the empty spaces would be the next solved number and the puzzle)

mark
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Based on the equation (9!^2(9)), the number of possible sudoku puzzles is exactly 1,185,137,049,600.

user293213
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