Hidden Pair

Hard

Description

Select a row, column or 3x3 block. If two missing digits can each go in only the same two cells, those cells must contain the two digits, one in each cell. You can rule out every other digit from those two cells. This is a Hidden Pair.

The group needs both digits, and neither can go anywhere else in it. They therefore occupy both cells, even though you do not yet know their order. The pair is called hidden because other candidates can obscure it in those cells.

You can use this reasoning without writing notes. Each example below first reserves two cells for a pair, then uses that restriction to find a Hidden Single of a different digit.

Example 1. A Hidden Pair in a block

In block 9, the existing 1s in row 7 and column 9, and the existing 6s in columns 8 and 9, confine both digits to row 8, column 7 and row 9, column 7. Those two cells cannot contain 2. — The highlighted 1s rule out the other two empty cells of block 9. The highlighted 6s rule out those same cells. Only the two purple cells remain for both 1 and 6, so neither cell can contain 2 or any other digit.

In row 8, the existing 2 in column 4 rules out column 4. The Hidden Pair rules out column 7. Only column 5 remains possible for 2. — Row 8 has three empty cells. The existing 2 in column 4 rules out one. The pair rules out column 7. Only the green cell at row 8, column 5 remains for 2 in the row.

Example 2. A Hidden Pair in a row

In row 7, the existing 3s in column 1 and block 8, and the existing 9s in columns 1 and 4, confine 3 and 9 to columns 3 and 9. Row 7, column 9 cannot contain 1. — In row 7, the highlighted 3s and 9s rule out columns 1 and 4 for both digits. The two purple cells, at columns 3 and 9, must contain 3 and 9. In particular, row 7, column 9 cannot contain 1. The pair's cells need not be adjacent.

In block 9, the existing 1 in column 7 rules out row 9, column 7. The Hidden Pair rules out row 7, column 9. Only row 8, column 8 remains for 1. — Block 9 has three empty cells. The existing 1 in column 7 rules out one, and the pair rules out row 7, column 9. The green cell at row 8, column 8 must contain 1.

Demo

Click on arrow buttons or use arrow keys on your keyboard to navigate between slides. Tap on arrow buttons to navigate between slides.

How to find Hidden Pairs

Choose one row, column or block, and check where each missing digit can go.
Find a digit with exactly two possible cells. Check whether another missing digit has exactly the same two possible cells in that group.
Keep the two digits in both cells. Remove every other candidate from those cells.
Look again for Hidden Singles and Naked Singles. A removal can create a placement elsewhere.

Without notes, use existing digits in intersecting groups to rule out cells, as in the examples above. With notes, highlight one digit at a time and compare its positions with another digit. Keep all remaining candidates recorded: an accidentally missing note can make a false pair appear valid.

Check both digits in the whole group

Seeing two cells that both contain 4 and 5 is not enough. In your chosen group, neither 4 nor 5 may be possible in any other cell. One extra occurrence of either digit defeats the Hidden Pair argument. The demo's last check shows this trap.

The two cells must share the row, column or block you checked; they do not have to touch. If either digit has only one possible position, start with that Hidden Single instead. If both cells already contain only the pair, there is nothing more for Hidden Pair to remove.

Using pencil notes

A cell with notes 4, 5, 7, 8, 9 and another with 1, 4, 5, 8 can form a Hidden Pair of 4 and 5, provided no other cell in their group allows either digit. The notes do not have to match before the deduction. Afterwards, both cells contain only 4 and 5.

In the demo, the green candidates are the pair to keep; red candidates are the other digits to remove from its two cells. Yellow highlights let you see all occurrences of the digits under discussion. Applying the pair removes candidates. Placing the resulting Hidden Single is a separate step.

At Sudoku Mood, Hidden Pairs first appear in Hard puzzles, where visual hints show how existing digits confine the pair. On Expert and Extreme, hints work with pencil notes. The demo uses a separate Expert position for that part so there is no easier single, intersection, Naked Pair or Naked Triple available first.

Hidden Pair, Naked Pair, or pointing pair?

A Hidden Pair starts with two digits: each is confined to the same two cells in a group. Remove other digits inside those cells. A Naked Pair starts with two cells: each allows only the same two digits. Remove those two digits from other cells in the group.

The two views can describe the same restriction. In a group with four empty cells, a Hidden Pair leaves the other two missing digits for the other two cells, forming a Naked Pair there. With notes that alternative can be easier to spot.

A pointing pair counts two possible positions for one digit in a block, both in one line. A Hidden Pair concerns two digits. Compare Pointing Candidates for the different elimination rule.

Try it yourself

These puzzles open at prepared positions from our puzzle library. Simpler moves have already been made. Find the Hidden Pair, remove the other candidates from its two cells, then find the Hidden Single of a different digit. Use Hint to check the deduction.

Sudoku #1 — find a pair in a row, without pencil notes.
Sudoku #2 — find a pair in a block, without pencil notes.
Sudoku #3 — find a pair in a column, with prepared Expert pencil notes.
Sudoku #4 — find a pair in a block, with prepared Expert pencil notes.