Naked pairs: two cells, two digits

They are easy to check once you have pencil marks, and they remove candidates in places that singles cannot reach. Once you know the pattern, you will see it often.

What is a naked pair?

A naked pair is two cells in the same unit whose only candidates are the same two digits. Those two digits must occupy those two cells, one each. No other cell in that unit can hold either digit, so both digits can be removed from the rest of the unit.

How it works

1. Find an empty cell with exactly two candidates. Call the digits X and Y.
2. Look for a second cell in the same row, column or box with exactly the same two candidates.
3. The two cells must hold X and Y between them. Remove X and Y from every other empty cell in that unit.
4. Check for new singles. Some cells may now have only one candidate left.

The picture below shows the idea in row 4. Each cell is named with r and c, as explained in the first guide.

The picture shows the whole grid, with only row 4 filled in, because the pattern lives in that row. Cells r4c2 and r4c6 are marked x. Their candidates are both {3, 8}. Cells r4c1, r4c4 and r4c8 are marked -, because they all contained 3 or 8 as candidates before the pair was found. Their candidates are {2, 3, 8}, {3, 4, 8} and {7, 8}, so they lose 3 or 8 respectively.

How to spot it

• Start with two-candidate cells. Cells with only two candidates are the best places to look for pairs. Mark them first.
• Compare within a unit. Take each unit in turn. If two of its two-candidate cells show the same pair of digits, you have a naked pair.
• Use the same method for boxes. A pair inside a box is just as useful as a pair in a row or column, and it is easy to miss if you only look along lines.
• Keep your notes accurate. Hush’s notes clean themselves up when you place a number, so stale candidates don’t hide a pair.

A worked example

The row 4 picture comes from this situation. Row 4 has four givens: 5, 9, 1 and 6. The empty cells are r4c1, r4c2, r4c4, r4c6 and r4c8. The digits still missing from the row are 2, 3, 4, 7 and 8.

Suppose the candidates are:

• r4c1 {2, 3, 8}
• r4c2 {3, 8}
• r4c4 {3, 4, 8}
• r4c6 {3, 8}
• r4c8 {7, 8}

The cells r4c2 and r4c6 are a naked pair with {3, 8}. The digits 3 and 8 must go into those two cells, so they cannot go anywhere else in row 4. Remove them:

• r4c1 becomes {2}, so it is a naked single and 2 goes there.
• r4c4 becomes {4}, so 4 goes there.
• r4c8 becomes {7}, so 7 goes there.

One pair produced three singles. The two pair cells keep {3, 8}, and the rest of row 4 can be settled from there.

The same idea in a box

Pairs work inside a box too. The next picture is a different puzzle, showing only box 5, the center box.

The pair cells r4c4 and r6c6 are in the same box, but not in the same row or column. Both hold only {1, 7}, so 1 and 7 fill those two cells. The other empty cells of box 5 (marked -) lose 1 and 7. Two of them, r4c5 and r6c5, also share a row with a pair cell, and r5c4 and r5c5 share a column or the box with one. Whichever way you look, the pair removes those digits from the box.

Common mistakes

• Removing digits from the pair cells themselves. The two cells keep both digits. The removal applies only to the other cells of the unit.
• Treating almost-matching cells as a pair. {3, 8} and {3, 8, 9} are not a naked pair. The pair needs two cells with exactly the same two candidates.
• Mixing up units. The two cells must share a row, column or box. Two cells with the same candidates in different rows and columns, and no shared box, do not form a pair.
• Removing only one digit. Both digits of the pair must be removed from the other cells in the unit, wherever they appear.

Why pairs need pencil marks

Naked pairs are hard to use without candidates written down. A player who relies on memory will miss most pairs, because the two-candidate cells are easy to forget once the grid gets busy. The habit to build is simple: after a placement, look at the cells that changed, and update their candidates before you search for pairs.

It also helps to mark the two-candidate cells in a unit first. Those are the only cells that can start a pair, so they make a short list. Once a pair is found, check the whole unit again for singles, because removing the pair digits often creates one.

If you are new to the vocabulary, the how to solve sudoku guide explains units, candidates and singles from the beginning, and the Learn hub shows where naked pairs sit among the other techniques. For the most demanding puzzles, Expert sudoku explains how Hush’s hardest level builds on these same ideas.

Practice naked pairs in Hush

Naked pairs appear in Hush’s Medium level, together with locked candidates and hidden pairs. Medium puzzles are unlimited and free, and the daily puzzle is a Medium puzzle from Monday to Thursday. Hard puzzles extend the idea to naked triples and quads. Before that, read hidden pairs to see how a pair can be hidden among other candidates. For the earlier technique, see locked candidates. If you want to know how the levels are priced, Sudoku without subscription explains how Hush works.