Sudoku Strategy: Singles, Pairs and Pointing Explained

Good Sudoku solvers are not faster thinkers. They have a short list of things to look for, and they look for them in the same order every time. This guide gives you that list, starting with the two moves that solve every Easy puzzle and working up to the patterns found in Hard ones. If you are new to the game, read how to play Sudoku first.

One promise before we start: you never need to guess. Every puzzle on this site is tested by a solver that uses only the methods on this page. If it would need a guess, the puzzle is thrown away.

The Methods at a Glance

1. Only Place: Scan One Number at a Time

This is the bread and butter of Sudoku. Pick a number, say 3. Look at every 3 already on the grid. Each one "blocks" its whole row and its whole column. Now go box by box and ask where the 3 can still go.

Example. The center box has four empty squares. There is a 3 in row 4 (off to the left) and a 3 in row 6 (off to the right), so the top and bottom rows of the box are blocked. That leaves the middle row of the box, which has two empty squares, in columns 4 and 6. Column 6 has a 3 near the top of the grid. So the 3 goes in the middle row, column 4. It is the only place left.

The same idea works along a line. In a row with three gaps, ask of each missing number: which of these three squares can take it? Often the answer is just one.

2. Only Number: Count What a Square Can See

Now look at a single empty square and collect the numbers in its row, its column and its box. If eight different numbers are already there, the square must be the ninth.

Example. A square's row holds 1, 4 and 8. Its column holds 2, 6 and 9. Its box adds 3 and 5. That is 1, 2, 3, 4, 5, 6, 8 and 9: eight numbers. The square is 7.

This is harder to spot than "only place" because you have to look in three directions at once. Try it where the grid is crowded: squares that sit in a nearly full row and a nearly full column at the same time.

A Routine That Works

1. Fill the gaps. Any row, column or box with one empty square: write in the missing number.
2. Scan the numbers 1 to 9 in turn, box by box, looking for an only place.
3. Each time you place a number, look again at that same number. One new 3 often opens up the next 3.
4. When a full pass finds nothing, check the most crowded squares for an only number.
5. Still nothing? Start notes. The puzzle is asking for one of the methods below.

On an Easy puzzle you will never get past step 4.

3. When and How to Use Notes

Notes are small numbers that record what is still possible in a square. Don't start them too early. On an Easy puzzle they only clutter the grid. They earn their keep when scanning runs dry.

There are two ways to use them:

• A few at a time. When a number can go in exactly two squares of a box, note it in both. Later, when one of them is ruled out, the other is solved.
• The full set. Tap Fill In Notes and every empty square shows everything that could still go there. From then on the game keeps them tidy as you place numbers. This is the right tool for Hard puzzles.

With full notes on the grid, the two basic moves become easy to see: a square with a single note is an "only number," and a note that appears once in a row, column or box is an "only place."

4. Pointing: A Number Pinned to One Line

Sometimes you can't tell exactly where a number goes in a box, but you can tell which row or column it will be in. That is still useful news for the rest of that line.

Example. In the top-left box, the only squares that can take a 4 are both in row 2. You don't know which one it will be. But either way, row 2 gets its 4 inside the top-left box. So no other square in row 2 can be a 4. Rub the 4 out of your notes along the rest of row 2.

Now look at the top-right box. Suppose it had three squares that could take a 4, two of them in row 2. Those two are gone, and one square is left. You have placed a 4 that scanning alone could not find.

The mirror image works too. If the only squares in row 2 that can take a 4 all sit inside one box, then that box gets its 4 from row 2, and the other squares of the box can't be 4.

5. Pairs: Two Squares, Two Numbers

The plain pair

Find two squares in the same row, column or box whose notes are exactly the same two numbers, for example 2 and 8 in both. One of them will be the 2 and the other the 8. You don't know which way round, but between them they use up the 2 and the 8 for that row, column or box.

Example. Here are the notes in the five empty squares of a row:

Squares A and C are the pair. Take 2 and 8 out of the other three squares. Square B is left with only a 5, so it is solved, and placing it takes the 5 out of square D as well. One pair has turned a muddle of notes into a row that is nearly done.

The hidden pair

This is the same idea seen from the other side. Suppose that in one box the numbers 1 and 7 can each go in only two squares, and they are the same two squares. Then those squares must be the 1 and the 7, whatever other notes they are carrying. Rub out every other note in those two squares. The pair was there all along, hidden behind extra notes.

6. Hard Puzzles: The Same Ideas, Stretched

Hard puzzles on this site use a few more patterns. Each one is a bigger version of something you already know.

Triples

Three squares in a row, column or box that hold only three numbers between them, such as notes of 1 4, 4 9 and 1 9. Those three squares will use up the 1, the 4 and the 9, so take those numbers out of the rest of the line or box. Note that no single square needs all three numbers.

The rectangle (often called an X-Wing)

Pick a number, say 6. Suppose that in row 2 the 6 can go only in columns 3 and 8, and in row 7 the 6 can also go only in columns 3 and 8. Those four squares form the corners of a rectangle. The two 6s must sit on opposite corners, one in each column. Either way, columns 3 and 8 get their 6 from rows 2 and 7, so every other square in those two columns loses its 6.

The three-square wing

Take a square with two notes, say 1 and 2. It shares a line or box with a square holding 1 and 5, and another with a square holding 2 and 5. If the first square is 1, the second must be 5. If the first square is 2, the third must be 5. So one of those two will be a 5 for certain, and any square that lines up with both of them can't be 5.

You don't need to memorize these before trying a Hard puzzle. When you are stuck, tap Hint. It shows the pattern on the grid, marks the squares involved, and explains each step in a sentence. Seeing a pattern in your own puzzle teaches it better than any diagram.

Good Habits

• Prove it or leave it. If you catch yourself thinking "it's probably a 4," stop. Look elsewhere. A guess that turns out wrong can spoil twenty minutes of work.
• Keep notes honest. A missing note is worse than no notes at all, because pairs and triples depend on them. If in doubt, use Fill In Notes and let the game do the bookkeeping.
• After every number, check its neighbors. The row, column and box you just changed are the likeliest places for the next step.
• Change direction when stuck. Been scanning boxes? Try rows. Been chasing 1 to 9? Look for crowded squares instead.
• Take a break. Your puzzle is saved. It is remarkable how often the next number is obvious after a cup of coffee.

Moving Up a Level

Stay on Easy until you can finish without a hint. Then try Medium with the mistake marks switched off in Settings, so that you learn to check your own reasoning. When pairs and pointing feel routine, Hard is waiting. If you enjoy this kind of step-by-step thinking, FreeCell rewards the same patience with cards, and Mahjong Solitaire is a restful change of pace.