Learn · The maths
The numbers behind sudoku
You do not need any of this to solve a puzzle. It does explain why the game is the way it is — and why “just make a puzzle” is harder than it sounds.
How many completed grids are there?
6,670,903,752,021,072,936,960 — about 6.67 × 1021. That is the number of ways to fill a 9×9 grid so every unit contains 1–9 once, counted by Felgenhauer and Jarvis in 2005.
If you consider two grids identical when one can be turned into the other by relabelling digits, swapping rows/columns inside a band, or transposing, the number collapses to 5,472,730,538 essentially different grids (Russell and Jarvis, 2006). Still five billion.
Why 17 clues?
A puzzle must have a unique solution, or logic cannot work: with two solutions, every deduction is a coin flip at the end. The fewest clues that can still force a unique solution is 17. This was conjectured for years and finally proved exhaustively in 2012 by McGuire, Tugemann and Civario.
There are only 49,151 known 17-clue puzzles (Gordon Royle's collection) — and most of them are not especially hard to solve. Minimum clues and maximum difficulty are different questions.
Why uniqueness matters
A puzzle with two solutions cannot be solved by pure logic: at some point two completions are both consistent with everything you have deduced, and no technique can choose between them. That is why every puzzle on this site is checked for a unique solution before it is published — and why “I found a different answer” is never a bug in a well-formed puzzle.
Why some puzzles resist deduction
Solving techniques are not magic; each one is a proof of the form “if this cell were anything else, two cells in a unit would collide”. The ladder of techniques corresponds to proofs of increasing logical depth. A puzzle is hard when no shallow proof exists — when the only way forward is a long chain of “suppose this were 3, then …”.
In our sample of the hardest public tier, 100% of puzzles could not be finished by the nine single-step techniques. That is not a flaw in the puzzles; it is the definition of the tier.
How puzzles are made
- Generate a complete grid (randomised backtracking).
- Remove digits one at a time, checking after each removal that the solution is still unique.
- Score the result with the technique ladder.
- If the score is not in the target band, adjust the number of clues and try again.
That loop is why “generated” and “hand-made” puzzles are not as different as people assume: the interesting part is the grading, not the digging.
About our corpus
This site ships 960 curated puzzles and 893,916 in the public source we grade from. The source is dedicated to the public domain, so the puzzles are free to play, print and share.