How many Sudoku puzzles exist, and the 17-clue result

That search found no 16-clue puzzle. All of these figures concern classic 9×9 Sudoku, and they say nothing exact about the variants.

How many completed grids are there?

A completed grid is a full 9×9 board that obeys the rules. In 2005 the mathematicians Bertram Felgenhauer and Frazer Jarvis counted them with a computer and found 6,670,903,752,021,072,936,960, which is about 6.67 × 10²¹ [1][2]. They did not try every grid. They used symmetry to cut the work down, as explained on the page about Latin squares and the math behind Sudoku.

The answer depends on what counts as different. If you relabel the digits, swap bands of rows, swap rows within a band, or turn the grid, you get another valid grid that is, in a sense, the same one in disguise. When Ed Russell and Frazer Jarvis counted grids up to those symmetries, they found 5,472,730,538 essentially different grids [1]. The two counts answer two different questions, and a careful statement always says which one is meant.

How many puzzles are there?

A puzzle is a partly filled grid with exactly one solution, so the same completed grid can give rise to a very large number of puzzles by hiding different cells. Counting them is harder than counting grids. A minimal puzzle is one from which no clue can be removed without losing the unique solution. According to the encyclopedia article used here, the number of minimal 9×9 puzzles is not known precisely, but a statistical method combined with a puzzle generator gives an estimate of about 3.10 × 10³⁷ minimal puzzles, and about 2.55 × 10²⁵ once equivalent puzzles are merged, with a relative error of 0.065% [1]. These are estimates from sampling and not exact counts. This page has not opened the paper behind them, which the article attributes to Denis Berthier (2009).

Whatever the exact figure, the numbers are astronomically larger than any collection a site could store. That does not mean a stored pool is unlimited: this site keeps 1,000 stored classic puzzles at each of five levels, which is a tiny sample of what exists, and the way they are varied is described in how puzzles are generated and verified unique.

What is the minimum number of clues?

The question is: what is the smallest number of given digits that a puzzle can have and still be solved in exactly one way? If too few clues are given, the grid can be completed in several ways. Puzzles with 17 clues were known for years, and nobody found one with 16, so people conjectured that 17 was the minimum [3]. Proving it was another matter, because checking every possible 16-clue arrangement against every grid is far too large to do naively.

In 2012 Gary McGuire, Bastian Tugemann and Gilles Civario posted a paper, later published in the journal Experimental Mathematics, that settled it. They searched exhaustively for a 16-clue puzzle, found none, and so proved that the answer is 17 [3][4]. Their method turned the question into a hitting set problem, a hard problem that is one of Karp’s 21 classic NP-complete problems, and a standard backtracking algorithm for hitting sets would not have been fast enough even on supercomputers. To make the search possible they designed a new algorithm that enumerates hitting sets efficiently [3]. A report in MIT Technology Review described it as a year-long calculation [5].

What does a hitting set have to do with Sudoku?

The following picture is this page’s own plain-language explanation, not wording from the paper, which this page has not read in full. Take a completed grid. Some groups of its cells can have their digits swapped around to produce a different valid grid. A unique rectangle, the pattern described in the unique rectangles lesson, is the smallest example: four corners that can trade two digits. For a puzzle drawn from that grid to have only one solution, its clues must include at least one cell from every such swappable group, so that the swap is blocked. A set of clues that touches every group is a hitting set. Asking for the smallest puzzle then becomes asking for the smallest hitting set, and the search showed that none of size 16 works for any grid.

What does the 17-clue result say, and what doesn’t it?

• It says: no classic 9×9 puzzle with 16 or fewer clues has a unique solution. Every valid puzzle needs at least 17 clues.
• It says: 17-clue puzzles exist, and the encyclopedia article reports that tens of thousands of distinct ones are known [1].
• It does not say that 17 clues make a puzzle hard, or that fewer clues mean harder. Difficulty depends on the techniques needed, as explained in how Sudoku difficulty is graded.
• It does not say anything about other boards. The result is for the classic 9×9 grid with 3×3 boxes. The minima for killer, diagonal or irregular boards are different questions, and this page does not state them.

The report in MIT Technology Review also noted that the result is a computer proof and not an elegant argument, and called for a proof that explains why the minimum has to be 17 [5].

How do this site’s puzzles compare?

The classic puzzles stored here have between 21 and 30 clues, with averages of about 24 to 25 at every level, according to the site’s own data. They are not close to the minimum, and they are not meant to be. The generator removes clues in random order and keeps a removal only if the puzzle stays unique and can still be solved with techniques no harder than its target level, so it stops at a puzzle that is irreducible for that level and not at the smallest possible one. The result is that clue count tells you almost nothing about difficulty here, which is the point made in the technique frequency guide.

Why does uniqueness matter?

All of these counts depend on the idea that a proper puzzle has exactly one solution. It is what lets a solver deduce each digit without guessing, and it is what the 17-clue search had to test for every candidate. The same property is checked for every puzzle on this site before release. The TypeScript solver tutorial shows how a program counts solutions and stops at two to prove a puzzle is unique.

To read the story behind the puzzle, see the history of Sudoku. The history hub lists all four pages, and you can play a classic puzzle on the home page.

Sources

1. “Sudoku.” Wikipedia. https://en.wikipedia.org/wiki/Sudoku. Accessed 22 October 2026. The article cites the OEIS sequences A107739 and A109741 for the grid counts, and Denis Berthier (2009) for the estimate of minimal puzzles; those cited documents were not opened for this page.
2. Bertram Felgenhauer and Frazer Jarvis. “Enumerating possible Sudoku grids.” 20 June 2005. Read through a copy at https://www.scribd.com/doc/94832755/sudoku, and through the summary in “The Math Behind Sudoku”, Math Explorer’s Club, Cornell University, https://pi.math.cornell.edu/~mec/Summer2009/Mahmood/Count.html. Accessed 22 October 2026.
3. Gary McGuire, Bastian Tugemann and Gilles Civario. “There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem.” arXiv:1201.0749, first posted January 2012, version 2 dated 1 September 2013. https://arxiv.org/abs/1201.0749. Accessed 22 October 2026. The abstract was read; the full paper was not.
4. The same paper as published: Experimental Mathematics 23(2), 190–217, 2014, as cited in source 1. The journal article was not opened.
5. “Mathematicians Solve Minimum Sudoku Problem.” MIT Technology Review. https://www.technologyreview.com/s/426554/mathematicians-solve-minimum-sudoku-problem. Accessed 22 October 2026. Publication date not shown in the copy read.