Naked Pairs in Sudoku: How to Spot & Use Them

The naked pair rule

A naked pair needs three things:

1. Two cells are in the same row, the same column or the same box.
2. Each of the two cells has exactly two candidates.
3. Both cells have the same two candidates.

Two cells with only two possible digits will use both digits: one cell gets the first digit, and the other cell gets the second. You do not know yet which cell gets which digit. But you know that both digits are used inside the pair.

A row, a column or a box contains each digit only once. So no other cell of the shared row, column or box can hold either of the two digits. Remove both digits from all of those cells. The two cells of the pair keep their candidates.

The word naked means that the two candidates are alone in their cells and easy to see. In a hidden pair, the two digits share their cells with other candidates.

The rule in one sentence

Two cells, the same two candidates, one shared row, column or box: remove both digits from the rest of it.

A real example, step by step

This board shows the candidates that are left at this point: the digits that are still possible in each empty cell.

Find the pattern

Look at column 3. It has five empty cells: R2C3, R3C3, R4C3, R7C3 and R9C3. R4C3 has the candidates 2 and 4. R9C3 also has the candidates 2 and 4, and nothing else. The two cells are in the same column and have the same two candidates. This is a naked pair on 2 and 4.

Remove the candidates

One of the two cells will be 2 and the other will be 4. So the 2 and the 4 of column 3 are both inside the pair, and no other cell of column 3 can hold them. Two other cells of the column have these digits as candidates. R2C3 has the candidates 1, 3, 4 and 8: remove the 4. R7C3 has the candidates 2, 3 and 4: remove the 2 and the 4. The fifth empty cell, R3C3, has the candidates 1 and 8, so nothing is removed from it.

R4C3 is in box 4 and R9C3 is in box 7, and the two cells are in different rows. Column 3 is the only row, column or box that they share, so the pair removes candidates only in column 3. R4C1 also has the candidates 2 and 4. But R4C1 and R4C3 are the last two empty cells of row 4 and of box 4, so those two cells leave nothing to remove there.

Use the result

A naked pair usually does not tell you which digit goes in which pair cell. Its value is in the eliminations. Look at R7C3. Before the naked pair, R7C3 had the candidates 2, 3 and 4. Now the 2 and the 4 are gone, and 3 is the only candidate left in the cell. That is a naked single: place 3 in R7C3.

Rows, columns and boxes

A naked pair works the same way in a row, a column and a box. Pairs in a box are often the easiest to see, because the nine cells are close together.

Two cells can share a box and a row or column at the same time. Then you can remove the two digits in both. Always check this. In the example, the two cells share only column 3, so there is only one place to remove candidates. If the two cells share no row, no column and no box, there is no naked pair, even when their candidates are the same.

How to find naked pairs

You need pencil marks to find naked pairs. Without the candidates written in the cells, the pattern is very hard to see. Search in this order:

1. Write the candidates. Do this at least in the areas where singles give no more progress.
2. Look for cells with exactly two candidates. Only these cells can be part of a naked pair.
3. Look for a partner. For each of these cells, check its row, its column and its box for a second cell with the same two candidates.
4. Remove the candidates. Remove both digits from every other cell of each row, column or box that the two cells share.
5. Look for singles again. As in the example, the eliminations can leave a cell with only one candidate.

Common mistakes

• Cells that share nothing. Two cells with the same two candidates in different rows, columns and boxes are not a naked pair. You cannot remove anything.
• A third candidate. If one of the two cells has a third candidate, the pattern is not a naked pair. Check both cells carefully.
• Removing in only one place. If the pair shares a box and a row or column, players often remove candidates in one of them and forget the other.
• Incomplete pencil marks. A cell can look as if it has two candidates when it really has three. A false naked pair leads to wrong eliminations, and later the puzzle cannot be solved.

Hidden pairs, naked triples and what comes next

Look for naked pairs in medium and hard puzzles each time no single is left. They are common, and they are easier to find than the advanced techniques.

The hidden pair is the related technique. Two digits are possible in only two cells of a row, column or box, but those cells also contain other candidates. You remove the other candidates from the two cells, and a naked pair remains. Both techniques use the same logic, seen from two sides.

The naked triple uses the same idea with three cells and three digits. The three cells together contain only three different candidates, and each cell may hold two or three of them. For example, cells with the candidates 2 and 5, 2 and 7, and 5 and 7 form a naked triple on 2, 5 and 7.

Naked pairs work well together with pointing pairs and claiming, which look at one digit where a box and a line cross. When these give nothing more, the X-Wing is usually the first advanced technique to learn. The full list of sudoku techniques shows all of them in order.