To solve Sudoku, place the digits that are forced first: a digit with only one place left in a row, column or box, or a cell with only one digit it can take. When those run out, pencil in notes and look for patterns that rule digits out. Each digit you rule out brings the next forced placement closer. Six techniques do all of this, and every level on PeerPuzzles can be finished with them.
New to Sudoku? Read the Sudoku rules first. They walk through a small 4×4 puzzle and show the first two techniques on a full grid.
A routine that works
- Scan for only places. Take one digit at a time and check every box, then any row or column that’s nearly full.
- Check crowded cells. A cell whose row, column and box are already busy may have only one digit left.
- Write notes once nothing is forced.
- Look for patterns in the notes: pointing, claiming and pairs.
- After every digit you place, look again at its row, column and box, and rub it out of the notes there.
The techniques below go in the same order. The first two place digits. The other four only rule digits out, but each one is usually a step away from a placement.
1. The only place for a digit
Pick a row, column or box and a digit it’s missing. Rule out every empty cell that already sees that digit, through its row, column or box. If one cell is left, the digit goes there.
The rules page shows this for a box, which is where most people start. It works just as well in a row or column, and sometimes only the row finds it: here neither the 9’s box nor its column has narrowed it down yet. A row, column or box with one empty cell left is the simplest case of all. The missing digit goes in the gap. More on hidden singles.
2. The only digit for a cell
Turn it around and start from a cell. It shares a row, column or box with 20 other cells. If eight different digits are already among them, the ninth is the only one it can take.
This position comes from a Hard puzzle with 39 digits in. No digit has only one place left anywhere on the grid, so scanning finds nothing. But row 8 already has 1, 2, 4, 5, 6 and 7, and column 8 adds 3 and 9. That leaves 8. Easy puzzles rarely need this view; Hard ones often do. More on naked singles.
Notes: when the singles run out
Notes (or pencil marks) are the digits each empty cell could still take. To write them, go cell by cell and pencil in every digit that isn’t already in its row, column or box. In the game, turn on Notes and tap digits to add them small.
Two habits keep notes useful:
- Rub out as you go. When you place a digit, remove it from the notes of every cell in its row, column and box.
- Be complete. Leaving a digit out of a note can make a pattern appear that isn’t there. Most stuck Sudokus have a missing note somewhere.
Notes don’t place anything by themselves. They make the next four techniques visible.
3. Pointing: a box’s digit stuck in one line
If all the places for a digit in a box lie in the same row or column, then that row or column gets its digit from this box. No cell outside the box in that line can have it.
The top-middle box needs a 1, and it has three empty cells. Column 5 already has a 1 (in row 4), which rules out row 3, column 5. Both places left are in column 4. You don’t know which one yet, but column 4’s 1 is in this box either way, so you can cross 1 out of rows 7 and 8 of column 4. With that gone, row 7 has only one place left for its 1: column 6. More on pointing pairs.
4. Claiming: a line’s digit stuck in one box
This is pointing the other way round. If all the places for a digit in a row or column lie inside one box, then that box gets its digit from this line. No other cell in the box can have it.
Row 1 needs a 9 and has three empty cells. Column 4 already has a 9 (in row 9), so the 9 is in column 1 or column 3, and both are in the top-left box. The box’s 9 is therefore in row 1, and you can cross 9 out of rows 2 and 3 of the box. That leaves row 2, column 3 with only 5 in its notes. More on box/line reduction.
5. Naked pairs
If two cells in a row, column or box have the same two digits in their notes, and nothing else, those two digits belong to those two cells. You don’t know which way round yet, but no other cell in that row, column or box can have either digit.
Row 4’s notes read: column 2 is 6, 7 or 9; column 3 is 7 or 9; column 5 is 1, 2 or 9; column 7 is 1, 2, 3, 6, 7 or 9; column 8 is 7 or 9; and column 9 is 2, 3 or 9. Columns 3 and 8 form the pair. Crossing 7 and 9 out of the rest of the row leaves column 2 with only 6, and gives column 2’s 7 only one place left, in row 8.
The same idea works with three cells that share three digits between them, called a naked triple. Pairs are far more common, so look for them first. More on naked pairs.
6. Hidden pairs
If two digits each have only the same two places in a row, column or box, those two cells must hold those two digits. Anything else in their notes can go. The pair is “hidden” because the cells have other notes too.
Column 6’s notes read: row 3 is 1, 5, 8 or 9; row 4 is 1, 5 or 7; row 6 is 7, 8 or 9; row 8 is 1 or 5; and row 9 is 1, 5 or 7. The 8 and the 9 each appear only in rows 3 and 6. So row 3 loses its 1 and 5, and row 6 loses its 7. Now row 6’s 7 has only one place left: column 3.
Hidden pairs are the hardest of the six to spot. Scan one digit at a time and count its places; when two digits share the same two, you’ve found one. More on hidden pairs.
How far these techniques go
Every level on PeerPuzzles is graded by these techniques. Easy levels need only technique 1. Medium levels need technique 2, 3 or 4 at least once, but no pairs. Hard levels need a pair, and the six always finish them.
That isn’t true of Sudoku in general, where difficulty is usually set by how many digits are given. We ran 300 random puzzles at each of three counts through a solver that uses only these techniques, always trying the simplest first:
| Givens | Techniques 1 and 2 | Up to 4 | All six |
|---|---|---|---|
| 40 | 100% | 100% | 100% |
| 32 | 93% | 95% | 95% |
| 26 | 59% | 67% | 72% |
With 40 givens, the first two techniques always did. With 26, two puzzles can be worlds apart: some need nothing beyond singles, and more than a quarter need techniques beyond these six, such as triples and X-Wing. That’s why our levels are graded by the techniques they need rather than by how many digits they start with.
A Medium puzzle, solved step by step
This puzzle is typical Medium: almost all singles, with one stuck point.
Step 1: place every single
Scan digit by digit and keep going until nothing is forced. Every one of the first 17 digits comes from technique 1, the only place for a digit. Then it stops, with 49 digits in.
Step 2: pencil in the 4s
Rather than noting every cell, try one digit at a time. Row 1 has gaps in columns 3, 5 and 9, and column 9 already has a 4. So row 1’s 4 is in column 3 or column 5.
Now look at the bottom-left box. Its 4 can only go in rows 7 or 8 of column 3, so column 3’s 4 is down there (technique 3, pointing). That rules out row 1, column 3, and row 1’s 4 has one place left: column 5.
Step 3: singles again
That one 4 is all it takes. The remaining 32 digits are all singles again, each one opening the next.
When you get stuck
- Look where you last placed a digit. Its row, column and box have just changed. Most new moves turn up there.
- Go one digit at a time. Check where the 1s can go in all nine boxes, then the 2s, and so on. It’s slower than glancing around, but it doesn’t miss things.
- Check your notes. A missing note hides moves. A note you forgot to rub out creates false ones.
- Start with the two-note cells. Pairs, and many single placements, begin with a cell that has only two digits left.
- Don’t guess. There’s always a reason for the next digit. Finding it is the puzzle.