Naked Pairs in Sudoku
Naked pairs are a powerful elimination technique used in medium and hard sudoku. Once you understand them, you'll unlock deductions that would otherwise seem impossible using single-cell analysis alone. A single naked pair can collapse two stuck cells into naked singles instantly — and that cascade often continues further.
What Is a Naked Pair?
A naked pair occurs when two cells in the same row, column, or box both contain exactly the same two candidates — and only those two.
For example: if cell A contains candidates {3, 7} and cell B in the same row also contains exactly {3, 7} — those two cells form a naked pair. One of them must be 3 and the other must be 7. We don't know which is which yet, but we know for certain that no other cell in that row can be 3 or 7.
This lets you eliminate 3 and 7 from the candidates of every other cell in the shared row, column, or box.
Why Does This Work?
Because the two cells must together account for both digits, those digits cannot appear anywhere else in the group. If a third cell in the same row were also a 3, then when the naked pair resolves, one row would contain two 3s — a rule violation. Therefore, 3 and 7 can safely be eliminated from all other cells in that group.
The key insight is that you don't need to know which cell gets which digit. The mere existence of the pair is enough to guarantee neither digit belongs anywhere else in the group. This is a logical deduction from constraints alone.
How to Find Naked Pairs
- Fill in pencil marks (candidates) for all empty cells.
- Look for cells that contain exactly two candidates.
- Check if any other cell in the same row, column, or box contains exactly the same two candidates.
- If found — you have a naked pair. Eliminate both digits from all other cells in that shared group.
Naked pairs require pencil marks to find reliably. Trying to spot them mentally on harder puzzles is error-prone. Always fill in complete candidates before searching for pairs.
Example: Naked Pair in a Row
In a row, five cells are already filled (1, 7, 6, 4, 9), leaving four empty cells. After applying row, column, and box eliminations, the candidates are:
- C2: {2, 8} — column 2 already contains a 3 (above) and a 5 (below), eliminating both
- C4: {2, 5, 8} — column 4 already contains a 3 (above), eliminating 3
- C6: {2, 3, 8} — column 6 already contains a 5 (below), eliminating 5
- C8: {2, 8} — column 8 already contains a 3 (above) and a 5 (below), eliminating both
C2 and C8 both contain exactly {2, 8} — a naked pair (amber below). One of them is 2 and the other is 8; both digits are now claimed by this pair and cannot appear anywhere else in the row. Eliminating 2 and 8 from the other empty cells:
- C4: {2, 5, 8} → remove 2 and 8 → {5} — a naked single!
- C6: {2, 3, 8} → remove 2 and 8 → {3} — a naked single!
One naked pair — two naked singles resolved at once. This is the power of naked pairs.
Naked Pairs in Columns
Naked pairs work identically in columns. If two cells in the same column both contain exactly the same two candidates, those candidates are eliminated from every other cell in that column. The searching process is the same — scan for two-candidate cells and check for matches in the same column. Column naked pairs are just as common as row naked pairs, so always scan columns when you finish scanning rows.
Naked Pairs in Boxes
Naked pairs in 3×3 boxes follow the same logic, but the elimination zone is the box itself rather than a row or column. If two cells in the same box both contain exactly the same two candidates, those two digits can be removed from all other empty cells in that box. Box pairs are often easier to spot visually because the nine cells are clustered together.
In the example below, the center box contains 1, 2, 7, 8, and 9 in five of its nine cells, leaving four empty cells that must hold 3, 4, 5, and 6. Column 5 already contains both a 4 (row 1) and a 5 (row 2) — these eliminate 4 and 5 from the two empty cells in column 5, reducing them both to {3, 6}. That is a naked pair inside the box.
The other two empty cells (both in row 5) have their candidates shaped by their own columns. Column 4 already contains a 5 (row 7), so the column-4 victim holds {3, 4, 6}. Column 6 already contains a 4 (row 7), so the column-6 victim holds {3, 5, 6}. The naked pair {3, 6} eliminates 3 and 6 from both victims:
- Column 4 victim: {3, 4, 6} → remove 3, 6 → {4} — naked single!
- Column 6 victim: {3, 5, 6} → remove 3, 6 → {5} — naked single!
How to Spot Naked Pairs Faster
The bottleneck for finding naked pairs is scanning. Here are practical habits that speed up the process:
- Hunt two-candidate cells first. After filling in all pencil marks, make a mental note of every cell containing exactly two candidates. These are your only pair candidates. Any cell with three or more candidates cannot be part of a naked pair.
- Group by value, not by position. If you see a cell with {4, 7}, scan the rest of its row, column, and box for another {4, 7} cell. Don't scan all pairs blindly — match a specific value set.
- Check dense rows and boxes. Groups with many filled cells have few empty cells, which means fewer candidates and a higher chance of pairs. A group with only four empty cells is a good scanning target.
Naked Pairs vs. Hidden Pairs
A hidden pair is the complementary technique. It occurs when two digits can only appear in the same two cells within a group — but those cells may contain other candidates alongside them. You find hidden pairs by looking at the digits (where can this digit go in this group?), not the cells.
The result is similar: once you identify the hidden pair, all other candidates in those two cells can be removed, and the pair resolves. Naked pairs are found cell-first; hidden pairs are found digit-first. Both are worth scanning for on hard puzzles.
Naked Triples
The same logic extends to three cells. A naked triple occurs when three cells in a group collectively contain only three distinct candidates. The cells don't each need to have all three — any combination works as long as the union of candidates across all three cells contains exactly three digits.
For example: {1,2}, {2,3}, {1,3} is a naked triple using digits 1, 2, and 3. Eliminate 1, 2, and 3 from all other cells in the group. Naked triples are rarer than naked pairs, but the search method is the same — look for clusters of cells whose combined candidates form a small, closed set.
When to Use Naked Pairs
Naked pairs appear on medium and hard puzzles, typically after naked singles and hidden singles are exhausted. If you're stuck and pencil marks are in place, scanning for two-candidate cells and checking for pairs is a reliable next step. Even one naked pair found in a single row can cascade: the two naked singles it produces each update candidates in their rows, columns, and boxes — potentially creating further hidden singles or even another naked pair.
On expert and evil puzzles, naked pairs may not appear at all, or they may appear only after pointing pairs and box-line reduction have been applied first. Always exhaust simpler techniques before searching for pairs.