Naked pairs

On this site naked pairs first appear in Medium puzzles.

The idea in one picture

Imagine two cells in a row whose pencil marks both read 6 and 7, and nothing else. You do not yet know which cell is the 6 and which is the 7, but you know they are the two of them. A row holds each digit once, so the 6 and the 7 of this row are spoken for. Any other empty cell in the row that still lists a 6 or a 7 can lose that mark.

The diagram above is a real position from this site's Medium pool, drawn by the technique engine. Its caption says that in row 6, the cells r6c6 and r6c9 hold only the digits 6 and 7 between them, so those digits belong to those cells and can be removed from the rest of row 6. Notation is row then column, counted from the top left.

What to look for

• Cells with exactly two candidates. Naked pairs are built from these, so a notes-filled grid is where you spot them. Scan a group for two bivalue cells carrying an identical pair of digits.
• Pairs that sit in one shared group. Two cells with the same two candidates only form a naked pair if they are in the same row, column or box. Cells that merely look alike elsewhere do nothing.
• Other cells in the group that still list either digit. Those are your payoff: each stray 6 or 7 is a mark you can erase.

Why the deduction is safe

Two cells, two digits, and no other option for either cell. If either cell took a digit outside the pair, it would have to be a candidate, and it is not. So the two cells use up both digits of the pair between them. That means no third cell in the group can use either digit. The reasoning does not need to know which cell takes which digit, which is why the technique works even when the pair cannot be resolved immediately.

It is worth being clear that a naked pair is a statement about cells, not digits. The cells are "naked" because the two digits are visible as the entire candidate list of each. Its mirror image, where the two digits are tucked among other candidates, is the hidden pair.

Naked pairs in a box, a row and a column

The group can be any of the three kinds, and a pair can work in more than one group at once. Two cells in the same row that also share a box form a naked pair for both the row and the box, so the elimination can reach cells in both. That double reach is why a pair near a box edge often produces the most progress. When you spot a pair in a box, check whether the two cells also line up in a row or column; if they do, you get a second, free elimination.

Common mistakes

• Treating a pair with one different digit as a pair. The two cells must match exactly: 6/7 and 6/7. A cell holding 6/7 beside a cell holding 6/8 is not a naked pair, although it is the start of an XY-Wing.
• Eliminating outside the shared group. Remove the digits only from cells that see both pair cells through the same group. A cell that sees just one of them is unaffected.
• Eliminating from the pair cells themselves. The pair keeps its own two marks. Erase everything else, never the pair.
• Forgetting to update notes. After an elimination, rescan the group for new singles. A naked pair often leaves another cell with one candidate.

Where the site's grading needs it

Naked pairs belong to the second tier, together with hidden pairs, pointing pairs and box/line reduction. In the committed frequency data over 1,000 stored classic puzzles per tier, a naked pair was used in 0 Easy puzzles, 285 Medium, 464 Hard, 494 Expert and 673 Evil puzzles. Each puzzle is solved cheapest technique first and a technique counts if used at least once, so the figures show what the engine reached for, not the only route. The page on how difficulty is graded explains the tiers and their limits, and the frequency guide lists every technique.

Where it sits in the learning path

Naked pairs are the step between single-digit thinking and set thinking. A naked single is a set of one cell and one digit. A pair is two and two. A naked triple is three and three. The method is identical; only the size grows, and the cells in a triple do not each need all three digits, which makes triples harder to see. Master the pair and the triple is a small step.

In the variants

Because every extra group in a variant is another place where cells can share a pair, naked pairs appear more often on boards with more constraints. On the diagonal board a pair can live on a diagonal. On the anti-knight board the knight-move neighbours also lose the pair's digits when they see both cells. On the 6×6 board the groups are small enough that you can see pairs at a glance.

Check your understanding

• Two cells in a box have candidates 2/5 and 2/5. A third cell in the same box has 2/5/9. What can you remove?
• Two cells in a row have 3/4 and 3/4, but they sit in different boxes. Can you remove 3 and 4 from a third cell in the first cell's box?
• Three cells have 1/2, 1/2 and 1/2. What does that tell you?

The answers: remove the 2 and the 5 from the third cell, leaving 9; no, because that cell sees only one of the pair cells through the box; and the position is impossible, since three cells cannot all fit into two digits, so a mistake has been made earlier.

A worked walk-through

Follow the diagram step by step. Row 6 has several empty cells, and two of them, r6c6 and r6c9, are down to the same two candidates, 6 and 7. First confirm both cells are in row 6: they are, so the row is the shared group. Second, list the other empty cells in row 6 and see which still carry a 6 or a 7. Each of those marks can be erased. Third, look at what the erasures did: a cell that had three candidates may now have two, and a cell that had two may now have one, which gives you a naked single to place.

The chain of consequences is the real payoff. A naked pair rarely finishes a puzzle on its own, but it commonly unlocks a run of singles, because it removes two digits from many cells at once. If a pair produces no immediate single, rescan the neighbouring boxes anyway; a stray 6 or 7 removed from a row can leave a hidden single in a box that crosses it.

How the engine reports it

When you ask for a hint and a naked pair is the cheapest technique available, the third hint level shows the elimination in the same wording as the diagram caption: the two cells, their two digits, the group, and the cells that lose a candidate. Because hints are computed from the current board and never read from the stored solution, the pair you are shown is always one that is genuinely on your board right now. The engine reads only the digits you have placed and works out the candidates itself, so you do not need to have written any notes for the hint to find a pair.

Practice

Play a Medium puzzle on the main board with notes on. When the singles run out, scan for two matching bivalue cells in a row, column or box. If you get stuck, the level-two hint names the technique without making the move. A Hard puzzle is likely to need a naked pair or a hidden pair somewhere in the middle of the solve.

Previous: hidden singles. Next: hidden pairs. Back to the technique list.