Naked Pairs: A Deep Dive

Naked pairs are the first elimination technique most solvers learn and the one most solvers misuse. They are easy to spot and very easy to mistake for something else, which is why a deep dive is worth your time before they become reflexive.

The shape is precise. Two cells in the same row, column or box, with exactly the same two candidates. Not similar. Not overlapping. Identical. The candidate set has to be the same two digits, and no third.

Why the pair is locked

The unit needs all nine digits exactly once. Cells A and B between them must hold X and Y (in some order), because those are the only two options either cell has. The other seven digits cannot live in A or B. They live elsewhere in the unit.

Therefore X and Y cannot live anywhere except A and B inside this unit. Any candidate X or Y outside those two cells, within the same unit, is wrong and can be erased.

A worked example

Imagine a row where cells 3 and 7 both show candidates {2, 7}, and the other cells in the row have richer candidate sets. The pair is locked into columns 3 and 7. Now look at the rest of the row. Any cell that still lists 2 or 7 as a candidate can have them removed.

r1c1 {1,2,5,8}   r1c2 {1,5,9}   r1c3 {2,7}*
r1c4 {1,5,8,9}   r1c5 {1,2,7,8}*  r1c6 {1,5,9}
r1c7 {2,7}*    r1c8 {1,8,9}   r1c9 {5,9}

r1c1 {1,5,8}    r1c2 {1,5,9}   r1c3 {2,7}*
r1c4 {1,5,8,9}   r1c5 {1,8}     r1c6 {1,5,9}
r1c7 {2,7}*    r1c8 {1,8,9}   r1c9 {5,9}

Cell r1c1 lost the 2. Cell r1c5 lost the 2 and the 7. Whether that produces an immediate placement depends on the rest of the grid, but in most cases it does. Pairs cascade.

The three most common mistakes

First, treating a "near pair" as a pair. If one cell shows {2, 7} and the other shows {2, 7, 9}, you do not have a pair. The candidates must match exactly. The cell with three candidates is unrelated to the technique.

Second, forgetting to apply the eliminations. The pair itself does not place anything. The eliminations elsewhere do. If you spot a pair and just admire it, you have not used the technique.

Third, confusing naked pairs with hidden pairs. Naked pairs are about the cells: two cells that, between them, can only hold two digits. Hidden pairs are about the digits: two digits that can only go in two cells, even if those cells have other candidates listed. They look similar in print and behave very differently in scanning.

Where to look

Pairs hide in units with three or four empty cells. Once a unit has eight or more candidates left, pairs are usually buried in noise. Once it has only one or two, the puzzle is almost solved and pairs are rare. The sweet spot is somewhere in between.

Boxes are richer than rows or columns because the box-line overlap creates more constraint. Most pairs you find will be in boxes first, then rows and columns as the grid fills in.

When the pair finds nothing

If you find a real pair and no eliminations follow, the pair was already implicit in the grid. That is fine. Move on. Sometimes pairs are just there as scaffolding for a future technique, and they will pay off two placements later.

For the formal definition and edge cases, see the naked pairs strategy page. Then try a hard puzzle and look for the first pair after about ten placements.