Naked and hidden pairs and triples

If naked and hidden singles are the two moves that solve most easy Sudoku puzzles, naked and hidden pairs and triples are the moves that solve most mediums and a respectable share of hards. They're extensions of the same logic — same perspective shift, same constraint reasoning — applied to two or three cells at a time instead of one.

The structural similarity is part of what makes them satisfying to learn. Once singles feel natural, pairs and triples slot in without much new mental scaffolding.

Naked pairs

A naked pair is two cells in the same unit (row, column, or box) where the only candidates are the same two digits. For example, if cells A and B in row 4 both have candidates 7 and nothing else, the digits 3 and 7 must occupy those two cells in some order — one of them holds the 3 and the other holds the 7.

The useful consequence: no other cell in row 4 can contain a 3 or a 7, because A and B have already claimed those digits between them. Even though you don't know which of A or B holds which digit, you know neither digit can appear elsewhere in the unit. That eliminates 3 and 7 from every other cell's candidates in row 4 — and the eliminated candidates often surface naked or hidden singles in those cells.

Naked pairs work because two cells with two candidates are exactly enough cells to hold exactly those candidates. The same logic extends to triples, and the same constraint elimination follows.

Hidden pairs

A hidden pair is the unit-first version of the same idea. In row 4, suppose digits 3 and 7 have only two cells where they could legally go — and those two cells are the same two cells. Even if those cells have other candidates listed, the digits 3 and 7 must occupy them between them, which means every other candidate in those cells can be eliminated.

The pattern is exactly parallel to the naked-single-versus-hidden-single distinction. Naked pair: two cells with two candidates → those candidates can't appear elsewhere. Hidden pair: two digits with two candidate-cells → those cells can't contain other digits. Same logical move, opposite perspective.

Hidden pairs are easier to miss than naked pairs, because the cells often have other candidates listed that hide the pair structure. The trick to spotting them is the digit-first scan: pick a unit, pick a digit, see how many cells in the unit could hold it, then check whether the answer is exactly two — and whether another digit has the same two candidate-cells.

Triples (naked and hidden)

Triples are the three-cell version of pairs. A naked triple is three cells in a unit where the candidates between them are exactly three specific digits, with each cell containing some subset of those three. The classic shape is three cells with candidates 2, 3, and 3 — each cell has only two of the three digits, but between the three cells the digits 1, 2, and 3 are exhausted, so no other cell in the unit can contain those digits.

Hidden triples are the unit-first version: three digits whose candidate-cells in a unit are exactly the same three cells. Same logic as hidden pairs, one cell larger.

Triples are noticeably harder to spot than pairs in practice. The reason is that the candidate sets don't have to be identical across cells — they just have to share the same three-digit universe. Most experienced solvers catch triples by scanning for naked pairs, finding none, and then asking whether a slightly weaker pattern is present.

When each shows up

Easy puzzles rarely require pairs or triples — most of their moves are singles. Medium puzzles have at least one pair somewhere mid-solve, often more. Hard puzzles tend to require pairs early and triples in the back half, often as the bridge to the harder techniques (X-wings, swordfish) that come next.

A practical pattern for medium and hard puzzles: solve every single you can find, mark the most-constrained cluster of cells, then look for naked pairs first (easiest to spot), hidden pairs second, naked triples third, hidden triples fourth. The first one you find usually unlocks several other singles and pairs in the surrounding cluster, and the puzzle accelerates from there.

Pairs and triples don't ever stop being useful. Even on expert puzzles where exotic techniques surface late in the solve, the back half is usually a stream of pairs and triples cleaning up the remaining cells. They're the mid-level workhorses, and getting them automatic is what makes hard Sudoku feel less like a wall and more like a slightly slower medium.