Two cells that have already agreed

The short version

A naked pair is two cells in the same unit whose candidate lists are both exactly the same two digits. Not "both contain" — both are exactly those two digits and nothing else.

Because those two cells must hold those two digits between them, the digits are spent. Every other cell in the unit loses both.

The shape

Find a unit — a row, a column or a box. Find two cells in it. Both must read {x, y} and nothing more.

If one cell reads {4, 7} and the other reads {4, 7, 9}, there is no pair. The second cell has an escape route, so nothing forces it to take 4 or 7, so nothing is spent.

This is the entire condition, and it is worth stating plainly because most false patterns come from relaxing it by accident. Both lists, exactly two digits, exactly the same two digits.

Why it works

Two cells, two digits, and each cell can only hold one value.

Suppose the pair is {4, 7}. The first cell takes 4 or it takes 7. Either way the other digit has to go in the second cell, because the second cell has nowhere else to go. So between them, the two cells consume exactly one 4 and exactly one 7.

A row, column or box can contain each digit only once. Both copies are now accounted for inside the pair. Therefore 4 and 7 cannot appear anywhere else in that unit.

Note what the argument does not need: it never decides which cell takes which digit. The eliminations are valid before, and regardless of, that question.

A real board

Two cells in row 1, both with exactly two candidates:

• R1C7 reads {5, 6}.
• R1C9 reads {5, 6}.

Identical lists, same row, two cells. So one of them is 5 and the other is 6, and row 1 has now used both of its 5s and 6s.

The consequence is in the rest of row 1: R1C3 loses its 5. It was a candidate there, and it is not any more. Row 1 gets its 5 from one of the two pair cells.

Notice how cheap that was. Two candidate lists compared, one elimination, and the row is measurably closer to resolved.

How to scan for one

Naked pairs are found by looking at cells, not digits, which makes them a different sweep from the one you use for hidden singles.

1. Work unit by unit. Pick a row and read the candidate lists of its empty cells. You are looking for two lists that are identical and short.
2. Short first. A pair must have exactly two candidates, so ignore every cell with three or more. This cuts the work by most of the board.
3. Compare within the unit only. Two identical {5, 6} cells in different rows that share no unit form nothing. The justification is scoped to one unit.
4. Check all three unit types. A pair found in a box eliminates inside the box. If those two cells also share a row, the elimination applies in the row too — but only because they genuinely share it, not by assumption.
5. Look for the follow-up. After deleting, re-read the affected cells. A cell that drops to one candidate is a naked single, and that is where pairs usually pay off.

A useful shortcut: pairs cluster in congested units. If a row has many cells at three or four candidates and none of them are resolving, that row is where a pair is most likely hiding.

The three traps

The near-pair. {4, 7} and {4, 7, 9} look like a pair at a glance and are not. The extra candidate is the whole point.

The false trio. Three cells all reading {4, 7} is not a pair and not a triple. It is three cells that could be 4 or 7, which is one cell too many for the digits to be exhausted. Delete nothing.

The cross-unit pair. Two {4, 7} cells in the same row but different boxes eliminate along the row and nowhere else. Solvers routinely delete from the boxes as well, because the cells "look related". They are related only through the unit they actually share.

Naked pair vs hidden pair

These two are confused more than any other pair of techniques, and the confusion is directional.

• Naked pair: two cells are locked to two digits → remove those digits from the rest of the unit.
• Hidden pair: two digits are locked to two cells → remove the other candidates from those two cells.

In the naked case the cells are the evidence and the unit pays. In the hidden case the digits are the evidence and the cells pay. If you find yourself deleting from a cell because of a naked pair, you have the direction backwards.

Hidden pairs, worked through on a real board.

Where to use it

Naked pairs are the first technique that requires a full candidate grid, which makes them the natural companion to the moment a puzzle stalls. They appear constantly in medium and hard puzzles, and they are cheap to check once the candidates are written.

They also generalise directly: three cells sharing three digits is a naked triple, and the reasoning is identical, just wider. If the pair feels obvious, the triple is the next step and not a new idea.