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Sudoku Hidden Pairs: Two Digits, Two Places

A hidden pair is two digits that can only go in the same two cells of a row, column or box. Those two cells must hold those two digits, one each, so anything else in their notes can go. It’s “hidden” because the two cells still have other notes, so nothing about them stands out until you look at where each digit can go.

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This is the fifth of the Sudoku techniques, and the hardest to spot. It needs full notes in the row, column or box you’re checking.

The rule

  1. In one row, column or box, find two digits that each have exactly two places.
  2. If they’re the same two places, those cells hold those two digits.
  3. Cross every other digit out of those two cells.
Hidden pair: 8 and 9 in column 6 Column 6 with full notes. The digits 8 and 9 each have only two places, rows 3 and 6, so those two cells must hold them. Their other notes, 1 and 5 in row 3 and 7 in row 6, are ruled out. 1 5 8 9 1 5 7 7 8 9 1 5 1 5 7 4 6 2 3 6 7 2 9 8 6 1 4 7 5 6 2 1 2 5 6 4 7 2 9 8 4 7 2 6 4 2 6 5 2 4 3 6 8 2 3 6
Column 6: 8 and 9 only fit in rows 3 and 6, so those cells lose their other notes.

In a real puzzle

The Hard puzzle every technique page follows gets here after five note steps and a naked single. Column 2 has six empty cells and plenty of notes:

Hidden pair in the Hard puzzle: 2 and 3 in column 2 Column 2 with full notes. 2 and 3 each appear only in rows 7 and 8, so those cells lose their other notes: 1 and 4 in row 7, and 4 in row 8. 1 6 8 1 4 6 8 4 6 8 1 4 6 8 1 2 3 4 2 3 4 3 5 4 2 7 2 3 5 2 1 3 5 9 2 2 9 7 8 8 1 8 6 7 6 8 1 9 2 3 5 1 9
Column 2: 2 and 3 only fit in rows 7 and 8, so row 7 loses its 1 and 4, and row 8 loses its 4.

Row 1 is 1, 6 or 8; row 2 is 1, 4, 6 or 8; row 3 is 4, 6 or 8; row 4 is 1, 4, 6 or 8; row 7 is 1, 2, 3 or 4; and row 8 is 2, 3 or 4. The 2 and the 3 appear only in rows 7 and 8. So row 7 loses its 1 and 4, and row 8 loses its 4.

That’s the turning point of the solve. With the 1 gone from row 7, column 2, row 7 has only one place left for its 1. That starts a run of ten digits placed one after another, nine of them hidden singles, before the puzzle needs its notes again.

The same pair, seen from the other side

A hidden pair always has a partner. Column 2’s other four empty cells, rows 1 to 4, hold only 1, 4, 6 and 8 between them: four cells, four digits. That’s a naked set of four (a “naked quad”), and it rules out exactly the same notes. In any row, column or box, a hidden pair and a naked set made of the remaining cells are two views of one fact. When the pair is easier to see, use the pair.

How often you’ll need them

On PeerPuzzles, 17 of the 50 Hard levels need a hidden pair, and no Easy or Medium level does. In Sudoku generally they’re rare: of 216 random 26-given puzzles that these six techniques could finish, only 3 needed one. But when one is needed, nothing simpler will do, so it’s worth knowing what to look for.

How to find them fast

  • Count each digit’s places. Go through one row, column or box digit by digit. Any digit with exactly two places is a candidate; look for a second digit with the same two.
  • Use Snyder notes in boxes. Snyder notes mark digits that have exactly two places in a box. Two digits marked in the same two cells are a hidden pair. See tip 6.
  • Try it when pairs and pointing run out. It’s the last of the six techniques for a reason: check for the others first.

Next steps

Questions

What's the difference between a hidden pair and a naked pair?

A naked pair starts from cells: two cells have only the same two digits. A hidden pair starts from digits: two digits have only the same two cells. A naked pair clears its digits from other cells; a hidden pair clears other digits from its cells.

Why are hidden pairs so hard to see?

Because the cells are cluttered. Each one still has other notes, so nothing about the cells stands out. You have to look at where each digit can go, not at what each cell can be.

Is there a hidden triple?

Yes: three digits whose places in a row, column or box are the same three cells (or fewer, between them). Those cells keep only those three digits. They’re rarer still than hidden pairs.