Sudoku techniques: how to solve every puzzle

Technique 1 · basic

Hidden single

Pick a box and a number it lacks. Every copy of that number in the same rows and columns rules squares out. When only one square of the box can still take the number, it goes there, even if that square could hold other numbers too.

This is the move that solves most of an easy sudoku. It works the same along a row or a column: if a number has one place left in a line, that is its place.

Where to look: the numbers that are already on the board most often. Their last places are the easiest to find.

Technique 2 · basic

Naked single

Now turn it round: look at one square instead of one number. Count what its row, column and box already hold. When eight different numbers are there, the ninth is the only one left for the square.

Naked singles show up in crowded parts of the board, where a row and a column both run nearly full. With notes on the board, any square with a single note is one.

Where to look: squares where a nearly full row crosses a nearly full column.

Technique 3 · intermediate

Pointing

Inside one box, a number's notes may all sit in a single row or column. The box has to hold that number somewhere, and every option lies on that line, so the line's copy of the number is inside this box. Cross the number out of the rest of the line.

Pointing places nothing by itself, but it clears notes far from the box, and that often leaves a hidden single somewhere else.

Where to look: a box with few empty squares, and a number whose notes there all line up.

Technique 4 · intermediate

Box-line

The mirror of pointing. In a row or column, a number's notes may all fall inside one box. The line needs that number, so the box's copy is on this line, and the box's other squares can't hold it.

Look for it after a row or column fills up: its last few notes often crowd into one box.

Where to look: a row or column with three or four empty squares left.

Technique 5 · intermediate

Naked pair

Two squares in the same row, column or box that both hold the same two notes, say 1 and 3, use up those two numbers between them: one is the 1, the other the 3. No other square of that unit can be a 1 or a 3.

The same works with three squares that hold only three numbers between them, a naked triple. Pairs are far more common and easier to see.

Where to look: squares with exactly two notes. Check whether another square in their row, column or box has the same two.

Technique 6 · intermediate

Hidden pair

Two numbers that fit only the same two squares of a unit must fill those two squares. Whatever other notes the squares hold can go, which often turns them into a naked pair.

It is harder to see than a naked pair because the two squares are cluttered with notes. Count where each missing number can go in the unit; two numbers with the same two places give it away.

Where to look: a unit with many empty squares, after counting the places of each missing number.

Technique 7 · advanced

X-wing

Take one number. If it fits only two squares in a row, and only the same two columns in another row, the four squares form a rectangle. The two copies in those rows sit on opposite corners, either way round, so each of the two columns gets its copy from these rows. Cross the number out of the rest of both columns.

It works the other way round too: two columns with the same two rows. With three lines and three cross lines, the same idea is called a swordfish.

Where to look: a number with few notes left on a hard board. Look along its rows for those with exactly two places.

Technique 8 · advanced

XY-wing

Find a square with two notes, x and y, that sees one square with x and z and another with y and z. If the middle square is x, the first end is z; if it is y, the second end is z. Either way one end is z, so any square that sees both ends can't be z.

This is the hardest technique the puzzles here ever need. Look for it only when every easier move has run dry.

Where to look: squares with exactly two notes. Three of them in a bent line, sharing their numbers in turn, make a wing.

Common mistakes

• Writing a number you only think is right. A wrong number spreads: it rules out the right ones around it. If you are not sure, leave the square empty or write notes.
• Forgetting the box. It is easy to check the row and the column and miss the 3×3 box. Check all three every time.
• Notes that are out of date. A note left behind after you place a number elsewhere hides the next single. Tidy notes (in Settings) crosses them out for you.
• Hunting for hard patterns too early. Most stuck moments are a single you have not seen yet. Scan every number once more before you look for an X-wing.