Sudoku naked pairs and triples: the cell-set elimination

If two cells in the same row, column, or 3×3 box both have exactly the same two candidates and no others, those two digits must occupy those two cells — eliminate both digits from every other cell in the shared unit.

No knowledge of which cell gets which digit is required. You only need to know the two cells together hold the two digits, which is enough to remove both from elsewhere in the unit.

A row has nine cells and contains digits 1-9 exactly once. Suppose two cells in the row, call them cell A and cell B, both have candidate set {4, 7}. Neither cell has any other candidates.

Whatever ends up in A is either 4 or 7. Same for B. They’re different cells in the same row, so they must hold different digits. So between them, A and B hold both 4 and 7 — we just don’t know in which order.

That’s the lock. The 4 and the 7 are committed to this pair of cells. No other cell in the row can hold 4 or 7. Erase both as candidates from the other seven cells in the row.

The same logic works for columns and boxes — anywhere the two cells share a unit, the elimination applies.

On a Medium puzzle, suppose you’ve pencilled in candidates for row 6 and two cells both show only {3, 8}. Maybe one is in column 2 and the other in column 7.

You don’t need to figure out which one is the 3 and which is the 8. The naked pair tells you:

• Both 3 and 8 are locked into those two cells of row 6.
• Erase 3 and 8 as candidates from columns 1, 3-6, 8, 9 of row 6.

Often this elimination immediately exposes a naked single elsewhere in the row, which lets you place a digit. That placement frequently chains into one or two more. A single naked pair commonly drops 3-5 placements.

The same idea, with three cells and three digits. Three cells in the same unit, whose candidate sets are each a subset of some three digits {a, b, c}, together must hold those three digits.

The three cells don’t need identical candidate sets. The combinations that qualify are:

• Three cells with {a, b, c}, {a, b, c}, {a, b, c} — the obvious case.
• Three cells with {a, b}, {b, c}, {a, c} — three bivalue cells covering three digits.
• Three cells with {a, b}, {a, b, c}, {b, c} — mixed bivalue + trivalue, still covering only three digits.

In every case: the three cells contain a, b, c between them, so eliminate all three digits from the other cells in the shared unit.

Naked quads (4 cells, 4 digits) exist but are rare and tedious to spot manually. Most published puzzles don’t require them.

Naked pair: two cells whose candidates are LIMITED to two digits.

Hidden pair: two digits whose candidates are LIMITED to two cells — even though those two cells might also have other candidates listed.

Hidden pairs are harder to spot because you have to scan by digit rather than by cell. For each digit in a unit, check whether it appears in only two cells, then check whether the same is true for a second digit in the same two cells. If yes — those two cells are committed to those two digits; you can strip every other candidate from those two cells, exposing a naked pair.

Hidden pairs and naked pairs are dual techniques. Every hidden pair has a corresponding naked pair if you fill out the candidates fully — but you may spot one before the other depending on the candidate clutter on the grid.

Naked pairs become useful once you’ve filled in pencil marks. They’re invisible without pencil marks because the technique reads off the candidate count per cell.

The scanning protocol:

1. Pick a unit (row, column, or box).
2. List the empty cells with exactly 2 candidates. Mentally tag each by its candidate set, like {3,8}.
3. Look for two cells with the same tag. That’s a naked pair.
4. Erase both digits from the other cells in the unit.

On Medium and Hard puzzles you’ll usually find 2-4 naked pairs across the whole grid. Each one typically domino-chains 2-4 placements.

Naked pairs (and their hidden cousins) are the bridge from “solver who can do Medium” to “solver who can routinely finish Hard.” They’re also the prerequisite to spotting fish patterns:

• X-wings read off naked-pair-style logic applied across two rows or two columns.
• Swordfish + jellyfish are the 3-row and 4-row extensions.
• Hidden subsets use the same dual framing on cells with mixed candidates.

The mental habit naked pairs train — “count the candidates, look for symmetric collapses” — is the foundation for every advanced sudoku technique. Internalize this one and the next ones cost less to learn.