Naked and Hidden Pairs, Triples and Quads

You need full notes for everything here. If you don't write them yet, our pencil marks guide shows how. All diagrams are real positions from puzzles with exactly one solution, drawn after every simpler technique has been used. Rows are numbered 1 to 9 from the top, columns from the left, boxes reading like a book.

Naked pairs

If two squares in the same unit have exactly the same two candidates and nothing else, those two digits must go in those two squares, one each. We don't know which way round, but it doesn't matter: no other square in that unit can hold either digit.

Highlighted: 7 the pattern · struck through: 7 removed

In row 5, the squares at row 5, column 7 and row 5, column 8 both have exactly 8 and 9. They will use up the 8 and the 9 of row 5 between them. They also sit together in box 6, so the same applies there. That removes 9 from row 5, column 1 (in row 5) and 9 from row 4, column 7, row 4, column 8 and row 6, column 7 (in box 6).

When the two squares of a pair share a box as well as a row or column, always check both units. It is easy to clear the row and forget the box.

Hidden pairs

A hidden pair is the same idea seen from the digits' side. If two digits can each only go in the same two squares of a unit, those squares must hold those two digits. Any other candidates in those two squares can go.

Highlighted: 7 the pattern · struck through: 7 removed

In box 5, the digits 4 and 8 each have only two possible squares, and they are the same two: row 4, column 4 and row 5, column 6. So those squares are the 4 and the 8, in some order. Before this step their candidates were row 4, column 4 had 4, 7 and 8; row 5, column 6 had 4 and 8. Every candidate other than 4 and 8 goes, which removes 7 from row 4, column 4.

Hidden pairs are harder to see than naked ones, because the extra candidates hide the pattern. The way to find them is to ask, for each unit, "which digits have only two spots left?" and see whether two of them share the same spots.

Naked triples

Three squares in a unit whose candidates, taken together, are only three digits. Those three digits must fill those three squares, so the rest of the unit loses them.

The surprising part: no single square has to contain all three digits. Here is a real triple where none of the squares has all three.

Highlighted: 7 the pattern · struck through: 7 removed

In column 3: row 2, column 3 has 5 and 8; row 5, column 3 has 2 and 8; row 9, column 3 has 2 and 5. Only the digits 2, 5 and 8 appear across the three squares, so those squares will take the 2, 5 and 8 of column 3. That removes 2 from row 1, column 3 (in column 3).

Patterns like this are easy to miss, because no two of the three squares look like a pair. When three squares each have two candidates and the candidates chain round, as {5, 8}, {2, 8} and {2, 5} do here, you have a triple.

Hidden triples

Three digits that, in some unit, can only go in the same three squares. Those squares hold those digits, and their other candidates can be removed.

Highlighted: 7 the pattern · struck through: 7 removed

In column 3, the digits 3, 8 and 9 are only possible in the squares at row 3, column 3, row 4, column 3 and row 6, column 3. So those three squares are the 3, 8 and 9. Their other candidates go: 6 from row 3, column 3; 4 and 6 from row 4, column 3.

Quads

The same logic works with four squares and four digits. In practice, quads are rarely the step you need. In a batch of 1,076 puzzles our solver generated while writing these guides, a quad was the simplest available step only twice. If you find one, use it, but don't spend long looking.

A handy fact: in a unit with eight empty squares, a naked quad in four of them is always matched by a hidden quad in the other four. So if a big naked set is hard to see, look for its smaller hidden partner, and the other way round.

How to scan

1. Look for squares with exactly two candidates. Compare each with the other two-candidate squares in its row, column and box. Matching pairs are naked pairs.
2. For each unit, list the digits with only two possible squares. Two digits with the same two squares are a hidden pair.
3. Only then look for triples: groups of three squares whose candidates add up to three digits.

Where they come up

Naked pairs are one of the two techniques behind our hard puzzles, alongside locked candidates. Hidden pairs, triples and the patterns in our X-wing and XY-wing guides belong to expert puzzles. The techniques page lists them all in order.