How Many Sudoku Puzzles Are There?

There are 6,670,903,752,021,072,936,960 valid completed 9×9 sudoku grids, about 6.67 sextillion. Bertram Felgenhauer and Frazer Jarvis counted them in 2005. Many of those grids are the same grid in disguise (swap two digits, rotate, reflect), and once you remove those copies there are 5,472,730,538 essentially different grids, a result from Jarvis and Ed Russell. The number of possible puzzles is larger still, because each grid can be turned into a huge number of different clue sets. The fewest clues a puzzle can have and still have one solution is 17.

How the 6.67 sextillion grids were counted

Nobody listed every grid one by one. Felgenhauer and Jarvis split the problem. They first counted the ways to fill the top band (the first three rows) and found that the millions of possibilities fall into a much smaller number of classes that behave identically for the rest of the grid. They then used a computer to count how many ways each class could be completed, and multiplied back up. The whole computation ran in hours on a 2005 desktop. Their paper, "Mathematics of Sudoku I", appeared in Mathematical Spectrum in 2006, and Jarvis kept a summary page with the figures. Other researchers checked the number independently soon after.

For scale: if you solved one completed grid every second, getting through all of them would take about 200 trillion years, roughly 15,000 times the current age of the universe.

Why only 5.47 billion are really different

Take any finished grid and swap every 1 with every 2. The result is still a valid grid, but it is the same puzzle for a solver. The same goes for rotating the grid, mirroring it, swapping two rows inside the same band, or swapping two whole bands. Combine all of those moves and one grid can be turned into up to 1,218,998,108,160 others (9! digit relabellings times 3,359,232 rearrangements, as set out on Wikipedia's mathematics of sudoku page).

Dividing 6.67 sextillion by 1.22 trillion gives about 5.47 billion. The exact count is slightly higher than that simple division, because a few grids map onto themselves under some of the moves. Ed Russell and Frazer Jarvis handled those cases with Burnside's lemma and got exactly 5,472,730,538 (Russell and Jarvis). At one grid per second, that list would take about 173 years to go through.

Grids are not puzzles

A grid is the finished answer. A puzzle is a set of clues that leads to exactly one grid. Any grid can be the answer to a vast number of different puzzles, because you can remove clues in many different orders and still keep the solution unique. That is why there is no single agreed count of possible sudoku puzzles. The number is so large that it is only ever estimated, and even the estimates depend on definitions (for example, whether you count only minimal puzzles, where removing any clue breaks uniqueness).

So you will never run out. Sudoku Royale's puzzle library holds 457,048 graded puzzles, which sounds like a lot until you compare it with the 5.47 billion essentially different grids.

The 17-clue minimum

For years people found puzzles with 17 clues but never one with 16. Gordon Royle at the University of Western Australia collected tens of thousands of 17-clue puzzles, and the question of whether 16 was possible stayed open until 2012. That year Gary McGuire, Bastian Tugemann and Gilles Civario at University College Dublin published an exhaustive computer search. They checked every essentially different grid for a 16-clue set with a unique solution and found none (arXiv:1201.0749, later published in Experimental Mathematics, 2014). Their paper cites 49,151 known 17-clue puzzles in Royle's collection at the time.

Seventeen clues is a floor, not a guide to difficulty. Many 17-clue puzzles are quite easy, and some of the hardest known puzzles have 21 or more. We look at that in the hardest sudoku ever.

How many clues real puzzles have

Published puzzles sit well above the minimum. Across the 457,048 active, graded puzzles in Sudoku Royale's library (snapshot taken 24 September 2026), the fewest clues is 20 and the most is 47. The busiest clue counts are 24 and 25, which together make up about 37% of the library.

The 35 to 47 range is mostly our easiest pool. Harder levels use fewer clues on average, though clue count alone does not decide difficulty. What decides it is which techniques you need, covered in our guide to sudoku difficulty levels.

Related numbers

• Every sudoku grid is a Latin square with an extra box rule. Latin squares go back to Euler's work in the 1780s. More in the mathematics behind sudoku.
• Solving a generalized n²×n² sudoku is NP-complete, shown by Takayuki Yato and Takahiro Seta in 2003 (IEICE Transactions).
• A classic grid has 81 cells, 27 units and 20 "peers" per cell (the other cells sharing its row, column or box). See does sudoku go diagonal for the rule details.