There are 6,670,903,752,021,072,936,960 valid completed 9x9 sudoku grids, about 6.67 x 10^21. No puzzle can have fewer than 17 given digits and still have one solution, and a well-made puzzle always has exactly one. This page gives the figures, who proved them, and what our own catalogue shows about clues and difficulty. It suits players who are curious about the numbers behind the grid. (Americans search for "sudoku math"; in British spelling it is "maths".)
A "grid" here means a completely filled board that obeys the row, column and box rules. Bertram Felgenhauer and Frazer Jarvis counted them in "Enumerating possible Sudoku grids" (2005) and found 6,670,903,752,021,072,936,960.
Many of those grids are the same board in disguise. Relabel the digits (swap every 1 with every 2), rotate or reflect the board, or swap rows within a band of three, and you get a different-looking grid with identical structure. Ed Russell and Frazer Jarvis counted the grids left after removing those symmetries in "There are 5,472,730,538 essentially different Sudokus" (2006). The answer is the figure in the title: 5,472,730,538.
More than there are grids, and nobody quotes a single agreed number. A puzzle is a grid with some digits hidden, so one grid gives rise to a huge number of puzzles. Only the ones that still have exactly one solution count as proper puzzles, and that check has to be made puzzle by puzzle. For play, the practical answer is "far more than anyone can solve in a lifetime". Our own catalogue is a sample, not a count of the whole space: it holds up to 1,000 puzzles per level and mode, sorted by difficulty.
Before that, 17-clue puzzles were already known. Gordon Royle maintained a public collection of them, and none with 16 was ever found. The proof closed the question: 17 is the floor.
Seventeen is a mathematical limit, not a playing standard. Puzzles that thin are rare and mostly interesting to researchers. Our fewest-clue classic puzzle has 19 clues, and typical puzzles sit in the mid-20s (table below).
No. Any grid with digits removed is a "puzzle" in a loose sense, but many such boards have two or more valid completions. A properly made sudoku has exactly one, and that is the standard convention: Nikoli, the publisher that popularised the puzzle in Japan, required it (see the history of Sudoku).
Uniqueness matters for a practical reason. If a puzzle has one solution, every step of pure logic leads towards it, and you never need to guess. If it has two, a solver can be correct at every step and still finish on the "wrong" answer.
How we check it: every puzzle on this site is proven to have exactly one solution before it is published. The solver looks for a solution, then looks for a second one, and the puzzle is rejected if a second exists. Each puzzle is also rated by our port of Sudoku Explainer, which records the hardest logical step needed. The same uniqueness check covers Killer and 16x16 puzzles.
Classic sudoku uses no arithmetic. The digits are just nine different symbols: you could swap them for nine letters or nine colours and nothing changes. Solving is logic, the elimination of possibilities, so it is closer to a deduction puzzle than to a sum. That is why 16x16 boards can use A to G after 9.
The maths is behind the puzzle rather than in it: counting grids, symmetry, and proving minimum clue counts are combinatorics and computer search. The 17-clue result is an example of a hitting-set problem solved by exhaustive computation.
Arithmetic does appear in variants. In Killer Sudoku the digits in each cage must add up to a target, so you reason about sums and which digit combinations can produce them.
Not reliably. The clue count says how much you are told, not how deep the logic runs. Here are the clue counts of our own classic 9x9 puzzles by level, from our catalogue as of 29 September 2026:
| Level | Fewest clues | Average | Most clues |
|---|---|---|---|
| Easy | 30 | 34.5 | 36 |
| Medium | 22 | 27.8 | 40 |
| Hard | 22 | 27.0 | 44 |
| Expert | 21 | 24.7 | 29 |
| Master | 19 | 24.4 | 28 |
| Legendary | 20 | 25.3 | 31 |
Two things stand out. Average clue counts fall from Easy to Master, but Legendary averages more clues (25.3) than Master (24.4), even though it is rated harder. And a Hard puzzle can have 44 clues. Difficulty follows the hardest technique the puzzle forces, measured on the Sudoku Explainer scale, not the number of givens. See difficulty levels for what each band demands.
Killer makes the point sharper. Our Expert, Master and Legendary Killer puzzles have no given digits at all: the cages alone fix the solution.
If you want to feel the effect yourself, try a Legendary puzzle and see how few clues you are given, then how much logic they still require. The hardest Sudoku puzzles push that further. New to the rules? Start with how to play Sudoku.