Naked Pair - Breaking Through with Mutual Candidate Elimination
A Naked Pair occurs when two cells in the same unit share exactly the same two candidates, allowing those two digits to be eliminated from other cells in that unit. This article explains this essential intermediate technique for conquering Hard-level puzzles.
The Principle Behind Naked Pairs
A Naked Pair occurs when two cells in the same row, column, or block have exactly the same two candidate digits and no others. For example, if two cells in a row both have only candidates {3, 7}, then 3 and 7 must go in these two cells (though which goes where is undetermined). The crucial insight is that these two digits cannot appear in any other cell in the same row. Therefore, 3 and 7 can be eliminated from the candidates of all other cells in that row.
A Systematic Approach to Finding Them
Finding Naked Pairs requires accurate pencil marks as a prerequisite. The procedure is: (1) Identify all cells that have exactly two candidates. (2) Check whether any two cells in the same unit (row, column, or block) share the same candidate pair. (3) If a pair is found, eliminate those two digits from all other cells in the same unit. This elimination often reduces another cell to a single candidate (triggering a Naked Single), leading to a chain of further progress.
Extending to Naked Triples
The Naked Pair concept extends to three cells. A Naked Triple occurs when three cells in the same unit have candidates that are subsets of the same three digits. For example, if three cells have candidates {2,5}, {2,8}, and {5,8} respectively, they form a Naked Triple of {2,5,8}. These three digits cannot appear in any other cell in that unit.
When to Apply in Practice
The right time to look for Naked Pairs is after Naked Singles and Hidden Singles have been exhausted. When every cell has 2 or more candidates remaining and no cell can be confirmed, Naked Pair elimination becomes the breakthrough. Units with many cells having exactly two candidates are particularly likely to contain Naked Pairs.
Naked Pair vs. Hidden Pair
Beginners often confuse naked pairs and hidden pairs. A naked pair is when two cells hold only those two numbers as candidates, so the pair is visibly exposed. A hidden pair, by contrast, is when two numbers can go in exactly two cells of a unit, but those cells also contain other candidates, so the pair is concealed. The direction of elimination is opposite too. With a naked pair, you erase those two numbers from the other cells of the same unit. With a hidden pair, you erase the other candidates from those two cells themselves. The two are two sides of the same coin, so once you find one, making a habit of viewing the board from the other's perspective improves your spotting ability.
A Concrete Example and a Pitfall
Consider a concrete case. Of the nine cells in a column, two cells hold only {4,6}, and another holds {4,6,8}. By the naked pair, {4,6} is locked into the first two cells, so you can erase 4 and 6 from the {4,6,8} cell, fixing it as 8. The pitfall is mistakenly treating two cells as a pair when their candidates include more than 4 and 6. Cells holding {4,6} and {4,6,9} are not a naked pair; the cells must hold exactly the same two candidates and nothing else. Also, even if a pair holds, the board does not advance unless there are other candidates to eliminate, so check whether a new placement arises after the elimination.
Pencil-Mark Habits So You Never Miss a Naked Pair
Finding a naked pair depends entirely on the accuracy of your pencil marks. A missed or un-erased candidate can make you overlook a valid pair or misread an invalid one. In practice, it is vital to update the candidate marks for the affected row, column, and block every time you place a confirmed number. Lightly marking cells that drop to two candidates makes them easy to use as starting points for a pair search. On paper, writing each number in a fixed position within the cell (1 at top-left, 9 at bottom-right, and so on) makes cells sharing the same candidate pair line up visually, so naked pairs become far easier to spot at a glance.
Combining with Locked Candidates
The naked pair becomes more powerful when combined with locked candidates (a state where a number's candidates within a block are confined to a single row or column). For example, if in one block the candidates for the number 5 are limited to two cells in the same row, you can eliminate 5 from the other blocks in that row. The resulting reduction of candidates can satisfy the conditions for a naked pair elsewhere. From the midgame on, Sudoku is rarely solved in one stroke by a single technique; the board advances through a chain in which a naked pair elimination invites the next locked candidate, which in turn produces the next placement. An attitude of stacking each technique accurately ultimately leads to the shortest solving path. Rushing into guesswork tends to chain errors and force you to start over from the beginning.