Summary: After Naked Singles and Hidden Singles, the next scan that makes progress without guessing is a pair. A Naked Pair is two empty cells in the same row, column, or 3×3 box that hold the same two leftover digits. A Hidden Pair is two digits that have only two legal cells left in that house. Neither scan writes a digit. Each one strikes leftovers so a later single can place.
Start from singles, then lock two leftovers
Naked Singles and Hidden Singles place a digit. A pair does not. It proves that two digits belong to two cells, then removes those digits from every other empty in the same house — or, for a Hidden Pair, removes every other leftover from those two cells.
If you can still place a single, do that first. Use this page when two or more empties remain in a house and the same leftover keeps appearing. For the 9×9 rules and the first fills on a nearly-full box, see How Sudoku Works.
Count rows down from the top and columns left to right. A printed digit stays put. Blank cells are empty. In the figures, blue marks the cells a scan reasons from — the pair it locks — and amber marks what the scan achieves. Neither scan writes the digit that makes the pair.
Naked Pair: two cells, the same two leftover digits
A Naked Pair is two empty cells in one house whose leftover digits are the same two values. Those two digits must occupy those two cells — one each, in some order — so neither digit can live in any other empty of that house.
You find it by staring at two cells in one house. List the leftover digits in each empty. If two cells show exactly the same two leftovers, strike those two digits from every other empty in that row, column, or box. You do not yet know which cell gets which digit.
Worked example
Look at row 5. The blue cells are row 5, column 4 and row 5, column 6. The amber cell between them is row 5, column 5.
The blue cells stay empty in both halves: the pair locks two digits into them without saying which gets which. Only the amber cell gains a digit.
Row 5 already shows 3, 4, 5, 6, 7, and 8. Only three cells are empty.
Leftover digits in those three empties:
- Column 4 (blue) — row 5 already took 3, 4, 5, 6, 7, 8. Column 4 already shows a 9 at row 7. The center box already shows 7 and 8. Leftovers: 1, 2.
- Column 6 (blue) — the same row digits, plus the 9 at row 2 of column 6, plus 7 and 8 in the center box. Leftovers: 1, 2.
- Column 5 (amber) — the same row digits, plus 8 and 7 in column 5 (and in the center box). Nothing removes its 9. Leftovers: 1, 2, 9.
The two blue cells hold exactly {1, 2}. That is a Naked Pair in row 5. Those two cells must be 1 and 2, in some order, so neither 1 nor 2 can go in column 5 of this row. Strike 1 and 2 from that middle empty. The leftover there is 9 — the right-hand grid above.
The pair did not write 1 or 2. It only cleared the middle cell. That 9 is now a Naked Single: one cell, one legal digit.
Hidden Pair: two digits, only two homes in a house
A Hidden Pair is two digits that can go in only the same two empty cells of a single house, even when those cells still look like they could take several other digits.
You find it by staring at two missing digits in one house. Pick a house. Pick two digits it still needs. Cross off every empty that already sees one of those digits in its other row, column, or box. If both digits have only the same two homes left, those two cells must hold those two digits. Strike every other leftover from those two cells.
The cells are not a Naked Pair until you strip them. They still show extra leftovers. The pair is hidden among those extras until you ask where those two digits are allowed to live.
Worked example
Look at row 2. Ask where 8 and 9 can go in that row. The blue cells are row 2, column 4 and row 2, column 5.
One grid this time, with no amber: the scan places nothing. It locks 8 and 9 into the two blue cells and strips their other leftovers.
Row 2 already shows 1, 2, and 3. Its six empties are columns 1, 2, 4, 5, 6, and 8.
- Columns 1 and 2 already see 8 and 9 in the top-left box, so neither empty on the left of this row can be 8 or 9.
- Column 6 already sees 8 at row 5 and 9 at row 8, so that empty cannot be 8 or 9.
- Column 8 already sees 8 and 9 in the top-right box, so that empty cannot be 8 or 9.
The two blue cells are the only empties in row 2 that do not already see an 8 or a 9. Both 8 and 9 must live in those two cells.
Those cells are not a Naked Pair yet. Each still sees only 1, 2, and 3 in the row, so 4, 5, 6, 7, 8, and 9 all look possible if you only ask “what can this cell be?” The pair appears when you ask “where can 8 and 9 go in this row?” Strike 4, 5, 6, and 7 from the two blue cells. They now hold {8, 9} — still unplaced, but locked.
Which question are you asking?
- Naked Pair — Which two cells in this house show the same two leftover digits? Strike those two digits from every other empty in the house.
- Hidden Pair — Where can these two digits go in this house? If they share only two homes, strike every other leftover from those two cells.
A house with two empties and two missing digits is both at once: the leftovers on the cells are the pair, and those digits have only those two homes. Use the Hidden Pair scan when the two cells still show extra leftovers.
Scan singles first — they place a digit. When a house still has several empties and no single will fire, look for Naked Pairs (two cells), then Hidden Pairs (two digits). The same idea scales to three cells or three digits; finish pairs before you hunt those.
Do not guess. If three cells still accept the same two digits, that is not a pair yet. Leave it and look at another house.
Practice on a fresh 9×9
Work the two mini-grids above, then print a Hard or Very Hard 9×9 to practice Naked Pairs, or a Tough 9×9 to practice Hidden Pairs. Cover the solution, and place every Naked Single and Hidden Single you can see before you start hunting the pair. Normal is still a singles band. Hard and Very Hard can also come from many hidden singles and no pair — if a page never locks two leftovers, print the next level, and use Tough when you want a Hidden Pair.
Puzzle Maker Pro Sudoku prints classic 9×9 grids with a matching answer key. The generate walkthrough is Getting Started with Puzzle Maker Pro Sudoku. The free demo generates one page at a time if you want to try a board first.
FAQ
What is a Naked Pair in Sudoku?
Two empty cells in the same row, column, or 3×3 box whose leftover digits are the same two values. Those two digits must occupy those two cells, so you strike them from every other empty in that house.
What is a Hidden Pair in Sudoku?
Two digits that have only the same two legal empty cells left in a single row, column, or 3×3 box. Those cells may still look like they could hold other digits until you ask where the two chosen digits are allowed to go, then strip the extras.
Do I need pencil marks?
They help. A Naked Pair is two cells that show the same two leftover marks. A Hidden Pair is two marks that appear in only two cells of a house. You can also find both by listing leftovers house by house, as in the examples above.
Which scan should I do first?
Singles first — Naked Singles and Hidden Singles. If nothing places, hunt Naked Pairs (two cells with the same two leftovers), then Hidden Pairs (two digits with only two homes).
Does a pair place a digit?
No. A pair only removes leftovers. In the row-5 example the Naked Pair leaves a 9 in the middle cell; that 9 is a Naked Single, not part of the pair itself. In the row-2 example the Hidden Pair locks 8 and 9 into two cells without saying which cell gets which.
Where do I learn the 9×9 rules and the first singles?
How Sudoku Works for rows, columns, 3×3 boxes, and the first fills on a nearly-full box. Naked Singles and Hidden Singles for the two scans that place a digit before you need a pair.