Naked Pair in Sudoku: how to eliminate candidates

Naked Pair is one of the first intermediate techniques worth learning after Naked Single and Hidden Single.

Unlike basic techniques, Naked Pair does not always let you place a number immediately. Its main goal is to eliminate impossible candidates from other cells. Those eliminations can then create simple new moves, such as Naked Single or Hidden Single.

This technique is based on a very logical idea: if two cells in the same row, column or box can contain only the same two numbers, then those two numbers must occupy those two cells, even if you do not yet know in which order.

In this guide, we will see what a Naked Pair is, how to recognize it and how to use it correctly to move forward in the solve.

A Naked Pair is a pair of cells that belong to the same unit, meaning the same row, column or box, where both cells contain exactly the same two candidates.

For example, in a row there are two empty cells that can contain only 2 and 8. Neither cell has any other candidates. This means that 2 and 8 must necessarily occupy those two cells, even if you do not yet know which cell will be 2 and which will be 8.

The consequence is important: no other cell in the same row can contain 2 or 8 as a candidate. Those two numbers are already “reserved” for the pair.

Naked Pair does not place the numbers immediately, but it narrows the possibilities in the grid.

To have a true Naked Pair, three conditions are needed.

The first is that the two cells are in the same unit: the same row, the same column or the same box.

The second is that both cells have exactly the same two candidates. It is not enough for them to share two candidates if one of the cells also has others.

The third, more practical condition is to check whether those two candidates also appear in other cells of the same unit: only in that case does the pair produce useful eliminations.

For example, if two cells in the same row have candidates {4, 9} and no other candidates, then 4 and 9 must go in those two cells. As a result, you can eliminate 4 and 9 from the other cells in the row where they still appear as candidates.

A Naked Pair in a row is one of the easiest cases to recognize.

Imagine a row with several empty cells. Two of these cells have only candidates 1 and 6. Since both belong to the same row, number 1 and number 6 must be placed exactly in those two positions.

At that point, if other cells in the same row contain 1 or 6 among their candidates, those candidates can be eliminated.

This elimination might not complete the row immediately, but it can create new opportunities. For example, a cell that previously had candidates 1, 5 and 6 might be left with only 5, becoming a Naked Single.

The reasoning is identical in a column.

If two cells in the same column contain only candidates 3 and 8, those two numbers must occupy those two cells. You do not yet know whether the first will be 3 and the second 8, or the other way around, but you know that both are reserved for the pair.

As a result, you can eliminate 3 and 8 from the other cells in the same column.

A Naked Pair in a column can be a little less immediate to spot than one in a row, because vertical reading is often less natural. Highlights or ordered candidates help a lot.

Naked Pair can also appear inside a 3×3 box.

If two cells in the same box contain only candidates 5 and 9, those two numbers must be placed in those two cells. All other cells in the box can no longer contain 5 or 9 as candidates.

This case is very useful because boxes are compact areas and often full of constraints. Eliminating candidates from a box can quickly unlock other moves.

Sometimes a Naked Pair found in a box is also on the same row or the same column. In that case, the pair is valid in both shared units: you can eliminate the two candidates both from the other cells of the box and, if applicable, from the other cells of the row or column. The important thing is to eliminate only in the units that both cells of the pair truly share.

With a Naked Pair, you can eliminate the two candidates of the pair from the other cells in the same unit.

If the pair is in a row, you eliminate those candidates from the other cells in the row. If it is in a column, you eliminate them from the other cells in the column. If it is in a box, you eliminate them from the other cells in the box.

You should not eliminate candidates from cells that do not share the unit with the pair. This is a common mistake: seeing a pair and applying the elimination too far away.

The practical rule is simple: you can eliminate the candidates of the pair only from cells that share the same unit with both cells of the Naked Pair.

Imagine a row with these empty cells:

• cell A: candidates 2 and 7;
• cell B: candidates 2, 5 and 7;
• cell C: candidates 2 and 7;
• cell D: candidates 5 and 9.

Cells A and C form a Naked Pair because both have only candidates 2 and 7.

This means that 2 and 7 must occupy A and C. As a result, in cell B you can eliminate 2 and 7. Cell B is then left with the only candidate 5.

At this point B becomes a Naked Single and can be completed with 5.

This example clearly shows the value of Naked Pair: it does not always solve the pair directly, but it can unlock other cells.

The first mistake is treating two cells as a Naked Pair when they have only some candidates in common, but not exactly the same two candidates.

For example, a cell with {2,7} and a cell with {2,7,9} do not form a Naked Pair. The second cell has one extra candidate, so the pair is not closed.

The second mistake is eliminating candidates from cells that do not belong to the same unit. A Naked Pair found in a box allows eliminations in the box, not automatically across the whole grid.

The third mistake is always expecting an immediate number. Naked Pair is often an elimination technique, not a direct placement technique.

Naked Pair occurs when two cells in the same row, column or box contain exactly the same two candidates.

In that case, those two numbers must occupy those two cells and can be eliminated from the other cells in the same unit.

It is a very important intermediate technique because it teaches you to reason not only about numbers to place, but also about candidates to eliminate. Using it well is a key step in moving from easy Sudoku to medium Sudoku.