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Hidden Pair

Two digits with nowhere else to go in a house own the two cells they share - and everything else pencilled into those cells is noise.

SubsetsMedium6 min readChapter 4 of the learning path
In one line

If two digits can only be placed in the same two cells of a box, row or column, those two cells hold them - so every other candidate in those cells can be crossed out.

Removes candidatesWorks in any houseNeeds pencil marks
Before thisHidden SingleNaked PairYou should already be comfortable pencilling in candidates.
the house we are working incells the pattern ownscandidates that must goOnly box one is pencilled in, to keep the diagrams readable.

How to spot it

A hidden pair is found by counting, not by looking. Pick a house and ask, digit by digit, where can this digit still go? You are hunting for two digits whose answer is the same two cells.

1

Choose a house and read its empty cells

Boxes are the easiest place to start - nine cells, all in view. This box has four empty cells and is missing the digits 1, 2, 4 and 7.

Four empty cells, four digits to distribute.
2

Count the homes of each missing digit

1 and 2 can go in three cells each. But 4 and 7 are each pencilled into only two cells - and they are the same two cells.

That is the whole pattern. Two digits, two cells, no other homes in the house.

Zoomed into the box. Both greens sit in the same two cells - nowhere else shows a 4 or a 7.
3

Claim the cells, then clear them out

One of those cells takes the 4 and the other takes the 7. You do not yet know which digit goes in which cell, and you do not need to: what matters is that nothing else fits in either of them.

The 2 pencilled into both cells is dead and can be struck out.

Two candidates marked for deletion.
4

Take the elimination, then look again

Both cells are now reduced to 47. No digit was placed - but the box has one fewer place for a 2, which is often enough to make a single appear somewhere else.

After the elimination. Two locked cells, two candidates gone.
Rule of thumb

If a hidden pair fires and nothing else moves, re-scan the same house before looking elsewhere. A house tight enough to hide a pair is usually hiding a second deduction too.

Interactive

Hidden pair in a box

Open in practice

Four phases, driven by the same solver output the in-game hint uses.

  1. 1Focus the house
  2. 2Frame the two cells
  3. 3Mark the candidates for deletion
  4. 4Apply the eliminations
Fixture hidden_pair · 001

Why it works

Every house holds each of the nine digits exactly once. Suppose you tried to put the 2 into the first of our two cells. The 4 and the 7 would then both need the second cell - and one cell cannot hold two digits.

Proof by contradiction
Give the grey 2 the first cell and both remaining digits are forced into the red one. The same argument applies to the other cell, so neither can hold anything but 4 or 7. No case analysis, no guessing - the elimination is forced.
1Before
Both cells still carry a 2.
2After
The 2s are gone; the pair is locked.

Naked or hidden?

The two techniques are the same fact read from opposite ends. A naked pair starts from cells that hold only two candidates. A hidden pair starts from digits that have only two homes. In our box, both are present - and both remove exactly the same candidates.

Hidden pair · 4 and 7

Found by counting digits. The pair is buried under a 2 in both cells - that is what makes it hidden.

Naked pair · 1 and 2

Found by reading cells. Two cells show nothing but 1 and 2, so the 2 elsewhere in the box has to go.

Common mistakes

  • Missing a third home. Check every empty cell in the house, including ones you have already narrowed down to two or three candidates.
  • Only scanning boxes. Hidden pairs in rows and columns fire just as often and are far easier to overlook, because the cells sit far apart.
  • Eliminating in the wrong place. The eliminations happen inside the pair's own two cells, not in the rest of the house. That is the opposite of a naked pair.
  • Expecting a placement. Nothing gets written in. If you were hoping to fill a cell, look for a hidden single instead.
One question

Which two digits form a hidden pair in this box?

Count the homes of each digit before you answer.

Recap

Look for
Two digits whose only remaining homes in a house are the same two cells.
Scope
Any box, row or column.
Result
Every other candidate in those two cells is eliminated. Nothing is placed.
Tell-tale
A house with few empty cells where one digit keeps showing up in only two of them.
If it fails
Try the naked pair, or size up to a hidden triple.
0 of 44 techniques understoodBack to libraryNext · Naked Triple
Hidden Pair - how to spot it and what it removes