Sudoku technique 5 of 12
Naked pair
Two cells in one unit with the same two candidates own both digits.
What is a naked pair?
A naked pair is two cells in the same row, column or box that have exactly the same two candidates and nothing else. Between them, those two cells will use up both digits, one each, even though you do not yet know which way round. So no other cell in that unit can have either digit, and you remove both from every other cell in the unit.
How to spot a naked pair
With your pencil marks filled in, look for cells that have exactly two candidates. When you find one, scan its row, column and box for another cell with the identical pair — same two digits, no extras. A cell holding 1 and 6 pairs with another 1-and-6 cell, not with one that holds 1, 6 and 9.
Then sweep the shared unit and cross both digits out of every other cell. If the two cells share a box as well as a line, you can clear both units. The idea extends to three cells sharing three candidates, a naked triple. Our solver never relies on triples, so no Easy, Medium or Hard puzzle in our game needs one; some Expert puzzles do.
A worked example
Position from a Hard puzzle dealt by our Sudoku generator, at the point where the game's own solver takes this step.
Cells are named by row and column: r4c1 is row 4, column 1. Rows count from the top, columns from the left, and boxes are numbered 1 to 9 reading like a book, so box 1 is top-left and box 9 is bottom-right.
This position comes from a Hard puzzle in our game, after singles have stalled and no pointing pair or box-line reduction is available. Look at row 9. Its first cell, r9c1, and its last cell, r9c9, both hold only 1 and 6. They are far apart, but they are in the same row, and that is all a pair needs.
One of them will be the 1 and the other the 6, so row 9 has no room for a 1 or a 6 anywhere else. Cross them out of r9c2, r9c4, r9c5, r9c7 and r9c8. Most of those cells still have several candidates afterwards, but r9c8 drops from 1, 6 and 9 to just 9. That is a naked single: r9c8 is 9.
Why it works
Row 9 must contain one 1 and one 6. The cells r9c1 and r9c9 cannot hold anything else, and they cannot hold the same digit, so between them they take the row’s only 1 and only 6. If a third cell in the row were a 1 or a 6, one of the pair would be left with no digit at all.
Which difficulty levels need it
Hard and Expert. Our solver tries a naked pair only after singles, pointing pairs and box-line reductions have nothing to offer, so whenever it uses one, nothing simpler would have done.
In 199 Hard puzzles we sampled, the solver used a naked pair in 40, and 40 of them could not be finished without it. Expert puzzles need this and more.
Related: Naked triple , Hidden pair .
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