The rules of Sudoku

One rule, applied three ways. This page is the complete version — what makes a grid valid, why every proper puzzle has exactly one answer, and the handful of assumptions that quietly break a grid.

The grid

A Sudoku is 81 squares in a 9×9 arrangement. Those same 81 squares are grouped three different ways at the same time, and every square belongs to exactly one group of each kind:

• Nine rows, running across.
• Nine columns, running down.
• Nine 3×3 boxes, marked out by the heavier lines.

Collectively those twenty-seven groups are called units. Two squares that share a unit are said to see each other, and that relationship is where every deduction in the game comes from.

The rule

Every unit holds the digits 1 to 9, each exactly once.

That is the whole rule. It has three useful readings, and good solvers use all three:

• No repeats. A digit cannot appear twice in the same row, column or box.
• Nothing missing. Since a unit has nine squares and nine digits, every digit must appear in every unit.
• One place each. If eight squares in a unit are settled, the ninth is forced — and so is the digit it takes.

Digits are labels, not quantities. Nothing in Sudoku adds up, and you could play it with nine colours or nine letters without changing a single deduction.

Exactly one solution

A well-formed Sudoku has precisely one completion. That is not a nicety; it is what makes the puzzle solvable by reasoning alone. If a grid had two valid completions, then at some point you would reach a position where two different digits both worked — and no amount of logic could choose between them. You would be guessing, and half the time you would be wrong for no reason you could have seen.

Every puzzle on this site is checked for uniqueness while it is being built. The generator starts from a complete, valid grid and removes clues one at a time, re-solving after each removal and putting the clue back if more than one completion has appeared. A puzzle that is not uniquely solvable is never shown to you.

Clues and how many there are

The filled squares you start with are the clues or givens. They are fixed: the game locks them, and no amount of tapping will change one.

It is a proven result that no Sudoku with a unique solution can have fewer than 17 clues. Puzzles at that floor are curiosities rather than good puzzles, though, and this site does not go there: the sparsest grade, Expert, keeps 22 to 27.

Clue count is a poor measure of difficulty on its own. A grid with 30 clues can fall out in two minutes if they are placed generously, and a grid with 30 clues can be brutal if they are not. That is why each grade here is defined by two things at once — a clue range and the hardest technique a solver should need:

Things that are not rules

Plenty of assumptions get attached to Sudoku that are not part of it:

• Diagonals do not count. The two long diagonals have no constraint in classic Sudoku. (There is a variant where they do — it is a different puzzle.)
• Nothing sums to anything. Rows do add up to 45, but that is a consequence of the rule, not a rule. It is occasionally a useful check and never a solving technique.
• Symmetry is not required. Many published grids place their clues symmetrically because it looks pleasing. It has no bearing on the solution.
• There is no fixed order. You may fill squares in any order at all. Some orders are far easier than others, which is what strategy is about.

How grids get broken

A grid that will not finish has usually been broken much earlier. The common causes:

• A digit placed on a hunch. One unproved digit poisons every deduction downstream of it, and the contradiction can surface twenty moves later in a completely different corner.
• Stale pencil marks. A candidate that should have been eliminated but was not will happily lead you to a wrong conclusion. Placing a digit here clears it from the notes around it automatically, which removes the commonest version of this.
• Scanning a unit incompletely. Missing one occurrence of a digit while scanning makes an impossible square look available.

If a grid contradicts itself, undo back to the last placement you can actually justify. Unlimited undo exists for exactly this.

House rules here

The rules above are Sudoku's. These are this game's, and they are the only additions:

• Three mistakes. A wrong digit stays on the board in red and costs one of three tries. The run ends when they are gone; the puzzle can be restarted with the same clues.
• A hint allowance. Hints reveal one correct square and are limited per game — five on Easy, three on Expert.
• A clock that pauses. It runs while you play, stops when you pause, and your best time is kept separately for each grade.