Sudoku · Rules guide
Sudoku rules: how to play
One rule stated three ways, the twenty squares each cell can see, how to use pencil marks without confusing yourself, and what actually makes one puzzle harder than another.
The short answer
- Every row, column and 3×3 box must contain the digits 1 to 9 exactly once.
- No arithmetic is needed — the digits are just nine distinct symbols.
- Every square sees twenty others, and every deduction follows from that.
- A correct puzzle has exactly one solution and never requires guessing.
- The minimum number of clues for a unique puzzle is 17, proved by computer search in 2012.
The rules of Sudoku
Sudoku has one rule, stated three ways: fill the 9×9 grid so that every row, every column and every 3×3 box contains the digits 1 to 9 exactly once. The printed digits are the clues, and they never change.
- No arithmetic is involved. The digits are just nine distinct symbols; you could use letters or colours and nothing would change.
- No guessing is required. A properly made puzzle has exactly one solution reachable by logic alone.
- Every puzzle has exactly one answer. If you find two, the puzzle is faulty.
That is the whole rulebook. Everything else people write about Sudoku — singles, pairs, X-Wings — is not rules but technique: named patterns of deduction that follow from those three constraints.
Rows, columns, boxes — and why that is enough
Each square belongs to three units: a row, a column and a box. A digit placed anywhere removes that digit as a possibility from all twenty other squares it shares a unit with.
Two immediate consequences give you the two basic solving moves:
- Naked single — a square where only one digit is still possible. Write it in.
- Hidden single — a digit that can only go in one square of a row, column or box, even though that square has other candidates. Write it in.
An easy puzzle can be solved entirely with those two moves. Everything harder is the same idea applied to groups of squares rather than single ones.
Pencil marks and how to work cleanly
Pencil marks are the small candidate digits written in a square. There are two schools:
- Full marking — write every candidate in every empty square, then eliminate. Systematic, slower to start, and essential for hard puzzles.
- Scanning only — mark nothing, look for singles by eye. Faster on easy puzzles and how most people play casually.
The practical advice: scan first, and only start marking when scanning stops producing anything. Then mark completely — partial marks cause more errors than no marks at all, because you trust them.
What makes a puzzle hard
Difficulty in Sudoku is not about the number of clues. A 30-clue puzzle can be much harder than a 24-clue one. What determines difficulty is which techniques the puzzle requires:
| Level | Techniques needed |
|---|---|
| Easy | Hidden singles, naked singles |
| Medium | Naked and hidden pairs, pointing pairs, box-line reduction |
| Hard | Triples, X-Wing, simple colouring |
| Expert | Swordfish, XY-Wing, chains, and forcing patterns |
| Extreme | Long chains, nets, and uniqueness arguments |
The clue-count facts
- The minimum number of clues for a unique solution is 17. This was proved by exhaustive computer search in 2012 by McGuire, Tugemann and Civario — no 16-clue puzzle with a unique solution exists.
- There are 6,670,903,752,021,072,936,960 valid completed grids, calculated by Felgenhauer and Jarvis in 2005. Ignoring symmetries, that is about 5.5 billion essentially different ones.
Variants worth trying
| Variant | The extra rule |
|---|---|
| Killer Sudoku | Dotted cages with a target sum; no digit repeats inside a cage. |
| Jigsaw / Irregular | The nine regions are irregular shapes rather than 3×3 boxes. |
| Diagonal (Sudoku X) | Both long diagonals must also contain 1–9. |
| Hyper / Windoku | Four extra shaded 3×3 regions must also contain 1–9. |
| Thermo, Arrow, Renban | Lines and shapes carry ordering or sum constraints. |
| Samurai | Five overlapping grids solved together. |
The variant boom of the 2010s and 2020s — driven by online puzzle setters and video solving — has produced puzzles with a handful of clues or none at all, where the constraint rules alone determine a unique answer. They are a different and often more elegant pastime than classic Sudoku.
Common questions
Do you need maths to solve Sudoku?
No. The digits are just nine distinct symbols — you could replace them with letters or colours and the puzzle would be identical. Sudoku is pure logic.
Can a Sudoku have more than one solution?
A properly constructed one cannot. If you find two valid completions, the puzzle is faulty — which also means "uniqueness" arguments are a legitimate advanced solving technique.
What is the minimum number of clues?
Seventeen. A 2012 exhaustive computer search by McGuire, Tugemann and Civario proved that no 16-clue Sudoku has a unique solution.
Is guessing ever necessary?
Not in a well-made puzzle. Every square can be deduced, although the hardest puzzles need long chains of reasoning that feel close to trial and error.
How many Sudoku grids are there?
6,670,903,752,021,072,936,960 completed grids, or about 5.5 billion once symmetries are removed. The number of puzzles is far larger still.
Does the number of clues determine difficulty?
No. Difficulty depends on which solving techniques are required. A 30-clue puzzle needing an X-Wing is harder than a 24-clue puzzle solvable with singles.
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