How Sudoku Works

One rule, applied three times over

A standard grid is nine cells by nine, divided by heavier lines into nine boxes of three by three. The constraint is that each of the digits 1 to 9 appears exactly once in each row, exactly once in each column, and exactly once in each box. Those are the same rule pointed in three directions, and there is no fourth. Because each digit appears once in a nine-cell unit and there are nine digits, every unit ends up completely full – which is why saying a digit may not repeat and saying every digit must appear come to exactly the same thing here.

The digits are labels, not quantities

Nothing in the puzzle uses the fact that 7 is larger than 3. Replace the numbers with nine letters, nine colours or nine shapes and the puzzle is unchanged, which is worth telling anyone who says they are no good at maths: there is no arithmetic in it anywhere. A consequence is that any finished grid can be rewritten in 362,880 ways just by deciding which symbol means what – nine factorial – and every one of those grids is the same puzzle wearing different clothes.

The diagonals are not part of the rule

A common and reasonable assumption, because the two long diagonals look as though they ought to matter. They do not, and most arguments about the sudoku rules turn out to be arguments about a variant somebody met first. There is a well-known one that adds the diagonals as extra units, and it is a genuinely different puzzle with different solutions; a standard grid will usually have digits repeating along a diagonal, and that is not an error. The same goes for the other family of variants that add cages, sums or comparison signs: each is a rule bolted on to the one above, and none of them is implied by it.

Two conditions that are rarely written down

The first is that a proper puzzle has exactly one solution. A grid with two is not a hard puzzle, it is a broken one, because a correct solver can reach an answer that disagrees with the published key. The second is that a proper puzzle can be finished by reasoning, without ever having to pick a candidate and see what happens. Neither condition is visible by looking at a grid, and neither is guaranteed by a puzzle appearing in print. Both are checked here before a board is drawn.

Seventeen is the floor

How few starting digits can a proper puzzle have? Seventeen. Thousands of seventeen-clue puzzles are known, and an exhaustive computer search published in 2012 established that no arrangement of sixteen clues anywhere in a nine-by-nine grid produces a unique solution. It is a rare case of a puzzle question with a settled, proven answer. It also has almost no bearing on daily play: puzzles printed for people rather than for record-setting sit in the twenties, and difficulty is not what the clue count is measuring.

How to play sudoku, in the order that works

Start with the digit that already appears most often on the grid, because it is the most constrained and the quickest to place again. For each box it is missing from, cross out the rows and columns that already contain it; if one cell survives, that cell is settled. Work through the digits before you work through the cells. When that stops producing anything, switch to reading individual cells and asking what is left for each – and only when both have dried up is it time to write candidates into the grid and start comparing cells against each other. The techniques page picks up from there, and an easy board is the place to try the first two steps on something that will definitely yield to them.