Sudoku Technique: Hidden Pairs and Triples

Hidden pairs are the close cousin of naked pairs. They're trickier to spot — but just as powerful — and they often unlock progress when nothing else seems to fit.

Where a naked pair finds two cells that obviously hold two candidates (because nothing else is in those cells), a hidden pair finds two cells that hold two candidates because those two candidates have nowhere else to go in the unit.

The Hidden Pair Rule

Example

Consider this row from a partially solved Sudoku. Filled cells are 1, 4, 7, 9 (in cols 1, 3, 4, 7). The other five cells carry the candidates shown:

Now ask the key question: where can each missing digit go in this row?

• 2: cols 2, 5, 8 (three cells)
• 3: cols 2, 5, 6, 8, 9 (five cells)
• 5: cols 6, 9 only (two cells) ←
• 6: cols 2, 5, 6, 8 (four cells)
• 8: cols 6, 9 only (two cells) ←

Look closely at 5 and 8. Each one can only land in cols 6 or 9 — the same two cells. That means cols 6 and 9 must contain 5 and 8 (in some order). Other candidates in those cells can't be the answer.

After the eliminations:

Notice the elegant aftermath: the hidden pair revealed itself as a naked pair. That's common — applying a hidden-subset elimination often turns the puzzle's state into something easier than where you started.

Why does it work?

Each row in a Sudoku contains every digit 1-9 exactly once. So digit 5 must appear somewhere in this row — and the only candidate cells for it are cols 6 and 9. Same for digit 8. That's two digits competing for two cells. By the pigeonhole principle, each cell gets one of them. No room for any other candidate.

Hidden Triples (and Quads)

The same logic extends:

Spotting a hidden triple is harder than a naked one because the cells themselves can look messy — they may carry many other candidates. The trick is to scan by digit and look for three digits that all collapse into the same three cells.

Naked vs. Hidden — same idea, different angle

Both techniques shrink down to the same conclusion: “these N cells contain these N values.”

• Naked subset: spot N cells whose candidate lists only contain N values total. Eliminate those N values from the rest of the unit.
• Hidden subset: spot N values that can only go in the same N cells. Eliminate every other candidate from those N cells.

They're the same fact viewed from opposite directions. Practiced solvers learn to switch between the two views automatically.

How to spot hidden pairs in practice

• Pick a unit (row, column, or box).
• For each missing digit in that unit, list which cells can hold it. A pencil-marked grid makes this fast.
• Look for two digits with the same two-cell candidate list. That's a hidden pair.
• Erase every other candidate from those two cells.
• Re-scan. The new state often unlocks naked singles or other hidden subsets.