There are 6,670,903,752,021,072,936,960 possible ways to fill a 9×9 Sudoku grid legally — roughly 6.67 × 10²¹, or about 6.7 sextillion. That figure, computed in 2005 by mathematicians Bertram Felgenhauer and Frazer Jarvis, counts every completed grid that satisfies Sudoku's row, column, and 3×3 box rules (people.math.sc.edu).
What Does "Sudoku Combinations" Mean?
When people ask about Sudoku combinations, they are asking a counting question: how many different completed Sudoku grids can exist? A completed grid is one where every row, every column, and every 3×3 sub-grid contains the digits 1 through 9 exactly once. The number of such grids is the total number of Sudoku combinations.
The exact answer is:
6,670,903,752,021,072,936,960
That is approximately 6.671 × 10²¹ — about 6.7 sextillion. To put it in perspective, that number is vastly larger than the number of grains of sand on all the world's beaches, and enormously larger than the number of stars in the observable universe (community.middlebury.edu).
How Mathematicians Counted the Sudoku Combinations
Counting every valid Sudoku grid by brute force is impossible in practice — there are simply too many to enumerate one by one. Felgenhauer and Jarvis solved the problem in 2005 with a clever reduction strategy:
- Fix the first band. They showed that any completed grid can be transformed (by relabelling digits and permuting rows, columns, and bands) into a standard form for the top three rows, drastically cutting the search space.
- Enumerate the reduced cases. With the first band fixed, they enumerated the remaining possibilities — about 3.5 billion arrangements to check.
- Run the computation. Using a standard PC, the exhaustive search took roughly seven months of computing time in the mid-2000s.
- Scale back up. Multiplying the reduced count by the number of equivalent transformations gave the exact total: 6,670,903,752,021,072,936,960.
Their result was later verified by independent computations, and it is now the accepted figure cited across combinatorics literature (gottwurfelt.com).
Essentially Different Sudokus: 5.4 Billion
The 6.67 × 10²¹ figure counts grids that differ by even a single digit. But many of those grids are really the same puzzle in disguise. If you:
- relabel the digits (swap all 1s and 2s, for example),
- permute rows within a band, columns within a stack, or the bands and stacks themselves,
- rotate or reflect the whole grid,
…you get a grid that is mathematically equivalent — the same puzzle structure wearing a different costume.
When these symmetries are factored out, the number of essentially different completed Sudoku grids drops to 5,427,730,538 — about 5.4 billion. Still enormous, but a tiny fraction of the raw total (community.middlebury.edu).
Sudoku Combinations vs Latin Squares
A Latin square of order 9 is a 9×9 grid filled with digits 1–9 such that each digit appears exactly once in each row and column — but with no 3×3 box constraint. Sudoku grids are a subset of Latin squares with the extra box rule.
There are about 5.5 × 10²⁷ Latin squares of order 9, so the box constraint eliminates the vast majority of them: valid Sudokus are more than 800,000 times rarer than Latin squares of the same size. That single extra rule is what makes Sudoku a puzzle rather than a triviality (astrophys-neunhof.de).
| Structure | Count (approx.) |
|---|---|
| Latin squares of order 9 | 5.5 × 10²⁷ |
| Valid completed Sudoku grids | 6.67 × 10²¹ |
| Essentially different Sudoku grids | 5.43 × 10⁹ |
| Minimum clues for a unique solution | 17 |
The 17-Clue Result
A related combinatorial milestone concerns puzzles rather than grids. A printed Sudoku puzzle is a partially filled grid; the question is how few given clues can still force exactly one solution.
In 2012, Gary McGuire, Bastian Tugemann, and Gilles Civario completed an exhaustive computer search proving that 17 is the minimum number of clues for a Sudoku puzzle with a unique solution. No valid 16-clue puzzle exists — with 16 or fewer clues, the puzzle always has at least two valid completions. The proof required about 7.1 million core-hours of computation (community.middlebury.edu).
Curiously, the number of known 17-clue puzzles is itself large: mathematician Gordon Royle maintains a collection of 49,151 essentially different 17-clue puzzles, and the total count is believed to be far higher.
Why the Number Matters for Players
These giant numbers are not just trivia — they shape the game you play:
- Every puzzle you solve is unique. With 6.67 × 10²¹ possible grids, no two randomly generated Sudokus will ever repeat in your lifetime.
- Puzzles are generated, not listed. No one could store a catalogue of all Sudoku grids — at roughly 80 bytes per grid, the full catalogue would need about 5.3 × 10²³ bytes, far beyond all the digital storage on Earth (astrophys-neunhof.de).
- Difficulty comes from clue placement, not the grid. All completed grids are equally "valid"; what makes a puzzle easy or fiendish is which cells are revealed and how the solver must reason through them.
If you want to sharpen your solving, our guide to Killer Sudoku covers the arithmetic variant, and how to play Plusword explores a different kind of daily puzzle logic.
What the Number Means for Puzzle Designers
The 6.67 × 10²¹ figure is not just trivia — it shapes how puzzles are made.
- Random generation is safe. With so many possible grids, a good generator will essentially never repeat a puzzle. Your daily app puzzle is almost certainly one nobody has ever solved before.
- Symmetry is a choice, not a limit. The count includes every possible grid, so designers can impose aesthetic constraints — rotational symmetry, pattern constraints — and still have astronomically many puzzles to choose from.
- Difficulty is designed, not found. Since all completed grids are equally valid, difficulty comes entirely from which clues the designer removes. Two puzzles can share a solution grid yet play completely differently.
- Uniqueness checking is essential. With this many grids, brute-force generation must verify each puzzle has exactly one solution — which is why reputable makers test every grid before publishing.
How many possible Sudoku combinations are there?
There are 6,670,903,752,021,072,936,960 possible completed 9×9 Sudoku grids — approximately 6.671 × 10²¹, or about 6.7 sextillion. This was computed by Bertram Felgenhauer and Frazer Jarvis in 2005.
Who calculated the number of Sudoku combinations?
Bertram Felgenhauer and Frazer Jarvis computed the exact figure in 2005 using an exhaustive computer search that took about seven months on a standard PC of the era (people.math.sc.edu).
How many essentially different Sudoku grids exist?
About 5,427,730,538 — roughly 5.4 billion. This counts grids as the same when they can be transformed into each other by relabelling digits, permuting rows/columns/bands, or rotating and reflecting.
What is the minimum number of clues in a Sudoku puzzle?
In 2012, McGuire, Tugemann, and Civario proved that no valid Sudoku puzzle with a unique solution can have fewer than 17 given clues.
Are Sudoku grids the same as Latin squares?
No. Sudoku grids are a special subset of Latin squares of order 9 with the additional constraint that each 3×3 box contains digits 1–9 exactly once. Valid Sudokus are over 800,000 times rarer than Latin squares of the same size.
Could a computer store every possible Sudoku grid?
No. Storing all 6.67 × 10²¹ grids would require roughly 5.3 × 10²³ bytes — many times more than all the digital storage capacity on Earth.
How many 9×9 Sudoku puzzles have 17 clues?
Mathematician Gordon Royle’s collection contains 49,151 essentially different 17-clue puzzles, and the true total is believed to be much larger.
Does the number of combinations affect Sudoku difficulty?
Not directly. All completed grids are equally valid; difficulty comes from which cells are given as clues and the logic required to fill the rest — not from which of the 6.67 × 10²¹ grids sits underneath.
What is a Latin square?
A grid where each symbol appears exactly once in every row and column. Sudoku grids are Latin squares with one extra rule: each 3×3 box must also contain every digit once.
Could a computer solve every possible sudoku?
Easily — computers solve individual puzzles in milliseconds. The 6.67 × 10²¹ figure counts how many grids exist, not how hard they are; no computer needs to enumerate them all to solve any single one.