Most sudoku puzzles are solved with two techniques: find a square where only one digit can go, and find a digit that has only one square left in a row, column or box. Easy and medium puzzles need nothing else. Harder ones add pairs (two squares that between them must hold two particular digits) and pointing digits, which let you rule a digit out of squares far from where you found it.
This is the order a solver reaches for them, and the order our puzzles are graded by: each level in Sudoku says which of these you will need.
First, pencil in what you know
Every technique works on candidates: the digits a square could still take, given what is already in its row, column and box. You do not have to write them all down (on an easy puzzle you will see singles without them), but on hard ones, pencil marks are what make the patterns visible. Switch Notes on, or hold Shift while typing a digit.
1. The only digit for a square (naked single)
Look at one empty square and cross out every digit already in its row, its column and its box. If only one digit is left, that is the answer. Squares in a crowded row that also sit in a crowded column are the place to look first.
2. The only square for a digit (hidden single)
Turn the question round: pick a digit and a box, and ask where in that box the digit can go. Every row and column that already holds the digit rules out its squares in the box. When one square is left, the digit goes there: even if that square could hold other digits as well. This is the technique people use most, and scanning the digit that is already on the board most often is the quickest way to find one.
3. Pointing digits (locked candidates)
Sometimes a digit cannot be placed yet but can be pinned down to a line. If, inside one box, every square that could take a 4 lies in the same row, then the 4 for that row has to be in this box, so no other square in that row, outside the box, can be a 4. The same works the other way round: if a row's only places for a 4 all sit in one box, the rest of that box cannot hold a 4.
4. Naked pairs
Two squares in the same row, column or box that each have exactly the same two candidates (say 3 and 8) must take those two digits between them, in some order. So no other square in that unit can be a 3 or an 8, and you can remove both from their candidates. The same idea extends to three squares sharing three digits.
5. Hidden pairs
The reverse: if two digits can each go in only the same two squares of a unit, those squares must hold those two digits, and any other candidates in them can go. Hidden pairs are harder to see than naked ones because the two squares still look crowded with candidates.
Which puzzles need which
| Level | Techniques needed |
|---|---|
| Easy | Singles, with plenty of givens to start from |
| Medium | Singles, but with fewer givens and fewer obvious places to start |
| Hard | Pointing digits and pairs, as well as singles |
| Expert | More than pairs: longer chains of reasoning, such as X-wings |
Every puzzle in our game is graded by running exactly these techniques on it. If singles finish it, it is easy or medium; if pairs and pointing digits are needed, it is hard; if they run out before the grid is full, it is expert.
Beyond pairs: the X-wing
Expert puzzles need patterns that span the whole grid. The best known is the X-wing: if a digit can go in only two squares in each of two rows, and those squares line up in the same two columns, then the digit must take one corner of that rectangle in each row, so it can be removed from every other square in those two columns. Spotting it is mostly a matter of looking along one digit at a time.
When you are stuck
- Go digit by digit through every box, looking for hidden singles. It is slow, but it finds most stalls.
- Update your pencil marks: a stale mark hides a single that is already there.
- Look for squares with only two candidates, and for pairs among them.
- Use one hint. It fills the square a solver would find next, so it shows you where to look as well.
New to the game? Start with sudoku for beginners. Curious why every puzzle has one answer? Read does every sudoku have one solution. Or go straight to Sudoku.