sudoku · 6 min read

Sudoku Technique: Naked Pairs and Triples

Master the grid through the absolute placement of non-repeating digits.

Once you're comfortable with naked singles and hidden singles, the next step in your Sudoku toolkit is the naked pair and its siblings (naked triples, quads). These are the simplest of the “subset” techniques, and they crack medium-difficulty puzzles wide open.

The idea is simple: when a small group of cells in a unit collectively contains exactly the same small group of candidates, those candidates belong to those cells — and to no other cell in the unit.

Prerequisite: pencil-marking candidates

Subset techniques only work after you've pencil-marked candidates for each empty cell. That is, in each blank cell, write small digits in the corner showing the values that are still possible after applying row/column/box constraints. Most printable Sudoku grids have space in the cell corners for exactly this.

The Naked Pair Rule

Example

Consider this single row from a Sudoku in progress. The filled cells (8, 1, 6, 4) have already been placed. The five blank cells carry pencil-marked candidates:

Look at the candidates carefully. The first cell (col 1) and the third cell (col 3) both contain exactly {2, 5} — no other candidates. That's a naked pair.

The naked pair {2, 5} occupies cols 1 and 3. Therefore 2 and 5 can be removed from every other cell in the row — col 4 loses them (struck through in red).

After the elimination, col 4 reduces from {2, 3, 5, 7, 9} to {3, 7, 9}:

Two candidates eliminated from col 4 — much easier to solve from here.

Why does it work?

Two cells with exactly two candidates between them can only be filled in two ways: either col 1 is 2 and col 3 is 5, or the other way around. Either way, both 2 and 5 are spoken for in this row. They cannot appear anywhere else in the row — to do so would require placing a third copy of 2 or 5, which Sudoku forbids.

Naked Triples (and Quads)

The same idea generalizes. A naked triple is three cells in a unit that together contain only three distinct candidate values (each cell may hold any subset of those three). Those three values must fill those three cells, so they can be eliminated from the rest of the unit.

Continuing our example: after the naked pair eliminations above, the three blank cells in cols 4, 6, 8 all hold subsets of {3, 7, 9}. That's a naked triple:

A naked triple {3, 7, 9} across cols 4, 6, 8. (In our row there are no other cells to eliminate from — but if the same cells lay in a column or box, the triple would clear those candidates from every other cell in that unit.)

Important nuance: the three cells need not all contain all three candidates. A valid naked triple could look like:

Cell A: {3, 7}
Cell B: {7, 9}
Cell C: {3, 9}

The combined candidate set is still exactly {3, 7, 9} — three cells, three values, locked in.

Naked Quads

Naked quads (four cells, four shared candidates) follow the same pattern but are rarer. By the time a quad shows up, easier techniques (singles, pairs, triples) usually solved the puzzle already.

How to spot naked pairs in practice

Look at cells with only 2 candidates. Scan each row, column, and box for cells that have exactly 2 pencil marks.
Compare those cells within the same unit. If you find two with the same two candidates, you've found a naked pair.
Apply eliminations. Erase those two digits from every other cell in the same row/column/box.
Re-scan. Removing candidates often unlocks naked singles or hidden singles in nearby cells.

Play sudoku