Hidden pairs

By Multiplayer Sudoku · Updated

The short answer

If two digits can only go in the same two cells of a row, column or box, those cells must hold those two digits, so every other candidate in those two cells can be removed. The pair is hidden because the cells still show other candidates until you clean them.

Hidden pairs are the mirror image of naked pairs. Instead of asking "what can this cell be?", you ask "where can this digit go?", the same flip that turns naked singles into hidden singles.

They are harder to see than naked pairs because nothing in the pencil marks stands out. You find them by counting where each digit can go.

A hidden pair

91351351524768742158631591981561563972512542564567967184595933635635676749182184925349764916835261275718462893193623689716845
  • Where the pattern lives
  • The pattern
  • Candidates removed
  • Solved as a result

In column 3, 4 and 9 each fit in only two cells, and they are the same two: R4C3 and R6C3. Those cells must take 4 and 9, so their other candidates can go. Now the 7 has just one place left in row 4: R4C4.

A hidden triple

1671245712456137123413478956178912789125128632746783245678248912675617145714526147894832693853487481314781791491476521689128912364138913856875893636897241689418971689189523689
  • Where the pattern lives
  • The pattern
  • Candidates removed
  • Solved as a result

In column 8, 2, 6 and 7 fit only in R2C8, R3C8 and R7C8. Three digits, three cells: those cells must take them, so their other candidates can go. That leaves R7C8 with only one candidate: 6.

A hidden quad

235815786384257857896894789152678678323568235658938712456814568245683583576478125789258145895792576589145783458145895791578932589145678145678456812562561581439456715698457179593256759324964584581
  • Where the pattern lives
  • The pattern
  • Candidates removed
  • Solved as a result

In column 5, 1, 4, 7 and 9 fit only in R4C5, R7C5, R8C5 and R9C5. Four digits, four cells: those cells must take them, so their other candidates can go. That leaves R4C5 with only one candidate: 7.

This position comes from a puzzle generated on this site, after every single has been filled in. Boxes are numbered 1–9 from the top left, row by row; R5C4 means row 5, column 4.

How to spot it

  1. 1Pick a row, column or box with several empty cells.
  2. 2For each missing digit, count the cells where it can still go. Note the digits that have exactly two spots.
  3. 3If two of those digits share the same two spots, you have a hidden pair.
  4. 4Remove every other candidate from those two cells. They are now a naked pair; if the two cells also share another row, column or box, the pair clears its digits there too.
  5. 5Look for singles in the unit afterwards.

Why it works

Every row, column and box needs each digit once. If one digit has only two possible homes in a unit and a second digit has the same two, the two digits need both cells between them: each takes one. Nothing else fits there any more.

Once cleaned, the two cells show only the pair, so a hidden pair always turns into a naked pair.

Common mistakes

  • Counting candidates in only part of the unit. A digit that also fits in a third cell anywhere else in the unit breaks the pair.
  • Mixing units: the two digits must be limited to the same two cells within one row, column or box.
  • Stopping too early. After cleaning the two cells you have a naked pair; if they also share another row, column or box, check whether it removes anything there.

Questions

Why are hidden pairs harder to see than naked pairs?

Because the two cells look ordinary: they still carry other candidates. You only find the pair by checking, digit by digit, where each one can go.

Is there a hidden triple?

Yes: three digits that fit only in the same three cells of a unit. They are rarer and harder to spot than hidden pairs, and the logic is the same.

How is a hidden pair different from a hidden single?

A hidden single is one digit with only one possible cell, so you can place it. A hidden pair is two digits sharing two cells: you can't place them yet, but you can clear everything else from those cells.

Try it on a real grid.

Hard Sudoku

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