Naked and Hidden Pairs

Pairs are the first genuinely deductive technique in Sudoku. They don't place a digit directly — they remove candidates elsewhere, and the removals create the singles that let you carry on.

Naked pairs

If two cells in the same unit have identical candidate lists of exactly two digits, those two cells will take those two digits between them. You don't know which goes where. It doesn't matter. Both are spoken for, so neither digit can appear anywhere else in that unit.

In the row above, two cells both read 4 and 7. One will be the 4 and the other the 7 — that's forced regardless of order. So every other cell in the row loses 4 and 7 from its candidates. Here that resolves three cells at once: one drops to a single 1, one to a single 9, one to a single 2.

This is the pattern to internalise: a pair rarely does anything on its own, and then does a great deal one move later.

Where to look

• Two-candidate cells are visually distinctive — scan for cells with exactly two pencil marks and check whether any share a unit with a twin.
• A pair can live in a row, a column, or a box. If both cells happen to be in the same box and the same row, the elimination applies to both units at once, which is twice the payoff.
• The same logic scales. Three cells sharing exactly three candidates between them form a naked triple; four cells sharing four form a quad. Triples are worth learning — they don't need all three cells to hold all three digits, just for the union of their candidates to be three digits total, so 12, 23, 13 is a valid triple.

Hidden pairs

The camouflaged version, and the harder one to spot. If two digits appear as candidates in only two cells of a unit, then those two cells must hold those two digits — and every other candidate in those two cells can be deleted.

Look at the box above for 2s and 6s. Both digits appear in only two cells. The box needs a 2 somewhere and a 6 somewhere, and there are only two homes for them, so those cells are claimed. The 1, 5 and 8 written in them are impossible and come out.

Note the direction of the deduction. A naked pair clears candidates from other cells. A hidden pair clears candidates from the pair's own cells. Beginners often learn naked pairs, decide pairs are handled, and never pick up the hidden version — which is the one that unlocks Hard puzzles.

A working routine

When singles have dried up, a reliable order is:

1. Sweep for two-candidate cells and check each for a naked-pair partner in any of its three units.
2. Take the units with the fewest empty cells and, for each, count how many cells list each missing digit. Any digit appearing in exactly two cells is half of a possible hidden pair.
3. Apply every elimination you find before looking for more — eliminations create singles, and a placed single changes the picture completely.
4. If nothing gives, move to pointing pairs and box/line reduction, which work between units rather than inside one.

Accuracy matters more here

Single-based solving is fairly forgiving: a stale pencil mark usually just hides a deduction. Pair-based solving is not. A candidate you forgot to erase can manufacture a pair that doesn't exist, and acting on it will delete candidates that were genuinely needed. The grid stays consistent for a long time before the contradiction appears, and by then it's unrecoverable without a restart.

Before you trust a pair, re-derive both cells' candidates from scratch. It takes ten seconds and saves the puzzle.