Two digits with only two homes

The short version

A hidden pair is two digits that have nowhere to go except the same two cells of a single unit. Since both digits must be placed and there are only two squares available, those squares are theirs. Every other candidate written in them is wrong and can be erased.

It rarely produces a number directly. What it produces is a cell that suddenly has two candidates instead of four, which is usually enough to break a stall.

The definition, precisely

Take one row, one column, or one box. Look at a single digit and count where it can legally go inside that unit. If it can go in exactly two cells, it is confined to those two.

Now do the same for another digit. If that one is also confined to the same two cells, you have a hidden pair.

Two conditions, both required:

1. Each digit has exactly two possible cells in the unit — one cell would be a hidden single, three or more is not confined.
2. The two digits share both cells. Two digits each confined to two cells, but different cells, is not a pair and gives you nothing.

Why the elimination is valid

This is a pigeonhole argument, and it is worth stating plainly because it explains why the move is safe.

Both digits must appear somewhere in the unit — that is the rule of sudoku. Each of them has only two available cells, and those cells are the same two. Two digits, two slots, no alternatives: the slots must be filled by those digits, in some order.

Once you accept that, anything else pencilled into those cells is impossible. The cells are already spoken for. You do not know which digit goes where yet, but you know with certainty that nothing else fits.

A real column

Here is column 4 of a live game. Three cells are empty, and only two of them decide the move.

The column already contains 5, 8, 7, 9, 3 and 1, so the missing digits are 2, 4 and 6.

Check where each one can go:

• 2 can go in any of the three empty cells.
• 4 can only go in R1C4 or R9C4.
• 6 can only go in R1C4 or R9C4.

Two digits, two cells. R1C4 and R9C4 must hold 4 and 6 in some order, which means the 2 pencilled into both of them is not just unlikely — it is impossible. Remove it.

With the 2 gone, the column reads cleanly: R5C4 is 2 — it never had anything else — and the other two cells are 4 and 6 in some order. Three unresolved cells became a fully described line without a single digit being placed.

How to spot one

The scan is digit-first, not cell-first.

1. Pick a unit you can read at a glance. A row with four or five filled cells works well; a nearly empty row has too many candidates to count reliably.
2. For each missing digit, count its possible cells. You are looking for exactly two. Anything with three or more is not confined and can be skipped immediately.
3. Compare the two-cell digits. If two of them point at the same pair of cells, that is your hidden pair.
4. Erase everything else in those two cells. Not in the unit — in the cells.

In practice you are watching for the same shape a hidden single makes, one step further out. A digit with one home is a single. Two digits sharing two homes is a pair. Three digits sharing three homes is a triple, and the same logic applies.

The direction confusion

This is where most people get it backwards, and the error is expensive because it looks plausible.

A naked pair looks at cells and eliminates outward. A hidden pair looks at digits and eliminates inward. Doing the reverse in either case produces eliminations that are not justified, and you will not notice until the board contradicts itself twenty moves later.

A useful sanity check: with a naked pair, the two candidate lists must be identical. With a hidden pair, the two candidate lists are usually different — that is exactly why the move is needed. In the example above, both cells hold 2, 4 and 6, but the pair is 4 and 6, and the 2 is the thing being removed.

Why it matters more than it looks

Hidden pairs show up in the middle of a puzzle, precisely where nothing else works. Every cell has three or four candidates, no singles exist, and the board feels locked.

At that point, a hidden pair does two useful things at once. It shortens two candidate lists, and it often exposes a naked single immediately afterwards, because a cell down to two digits in a unit that already contains one of them becomes a naked single on the spot.

It is also a prerequisite for harder techniques. X-Wing, skyscrapers and forcing chains all assume the candidate lists are trustworthy. A missed hidden pair means those lists are too long, and long lists hide patterns.

When it does not apply

A digit can also go in a third cell. Then it is not confined, and there is no pair. This is the failure mode people miss most often, because the two cells still look neat and symmetric.

The two cells do not share the unit you are working in. A row-based pair says nothing about the column, even if both cells sit in the same column as something else. The justification only covers the unit you found it in.

The two cells are in the same box as well. Then you get a bonus: the elimination also applies inside that box, because the same two cells are in it. This is common and easy to forget.

You have not updated candidates after a placement. A hidden pair found on stale pencil marks is a mirage. If you filled a digit three moves ago and did not clean the affected cells, your counts are wrong and the pair may not exist.