Sudoku

The rules

A sudoku is a 9×9 grid divided into nine 3×3 boxes. Some squares already hold digits, the clues or givens. Fill in the rest so that every row, every column and every box contains the digits 1 to 9 exactly once. That is the whole rule set. You never add anything up; the digits are just nine different symbols.

A proper sudoku has exactly one solution, so every square can be worked out by reasoning. Every puzzle on this page is checked for that before you see it: a solver searches for a second solution and the puzzle is thrown away if one exists.

Solving techniques, easiest first

Singles

The simplest move: a square where only one digit is still possible. The easiest case is a box, row or column with eight digits already in it. Here the box has every digit except 6, so the empty square is 6:

More often you check a square against its row, its column and its box together. If eight different digits appear among them, the ninth is the answer. Easy puzzles here can be finished with this move alone.

Hidden singles (cross-hatching)

Turn the question around: instead of asking what goes in a square, ask where a digit can go. Below, the 5s in the other two boxes rule out the top two rows of the first box. In its bottom row two squares are taken, so the 5 has only one place left, even though that square might look open to several digits:

Scan each digit from 1 to 9 across the boxes this way; it finds most of the digits in medium puzzles.

Pointing pairs and box-line reduction

When a digit can only go in squares of one row inside a box, it must be in that row within that box, so it can't appear anywhere else in that row. Here the 7 of the first box is in one of two squares of the middle row, which rules 7 out of the rest of the row:

Box-line reduction is the mirror image: if a digit's only places in a row (or column) all sit inside one box, it can't be anywhere else in that box. Both need pencil marks, which is what makes hard puzzles hard.

Naked and hidden pairs

If two squares in the same row, column or box can only be 3 or 8, those two digits are used up between them, and no other square in that unit can be 3 or 8:

A hidden pair is the reverse: two digits that fit only the same two squares of a unit. Those squares can then hold nothing else, so their other pencil marks go. Use Fill in pencil marks on the board to see these patterns without writing every candidate yourself.

What the levels mean

Each puzzle is graded by a logical solver that works like a careful person: it always takes the easiest move available and records the hardest one it needed. The clue counts below come from 40 puzzles per level made by this page's generator:

Notice that hard and expert puzzles give about the same number of clues: what matters is where they are. In the sample, a medium puzzle needed about 8.5 hidden singles on the way, and 95% of hard puzzles needed at least one pointing pair. On expert puzzles the same solver got stuck after filling about 47.4 of the 81 squares (givens included): from there you need a more advanced pattern, or patience. A hint is always available; when no simple move exists it simply gives you a square from the solution.

Tips for playing on screen

• Use the highlights. Selecting a square shades its row, column and box, and every copy of its digit, so gaps stand out.
• Watch the number pad. Under each digit is how many are still to place; when it says done, that digit is finished.
• Notes on hard puzzles. Pencil in the two or three places a digit could go in a box. Placing a digit removes it from the notes in its row, column and box automatically.
• Undo freely. Undo and redo go back as far as you like, so it is safe to try an idea and step back.

Prefer paper? Make a stack of puzzles with answer keys on the printable sudoku page. For a different kind of logic, try minesweeper or a round of solitaire, or make a word search from your own words.