Naked Triples Technique: Three Cells Locking Three Numbers
Naked Triples is an extension of Naked Pairs and an important intermediate Sudoku technique. The core concept is: when three cells in the same row, column, or box have candidates that are subsets of the same three numbers, these three numbers must be placed in these three cells, so they can be eliminated from other cells in that unit.
If three cells in a row, column, or box all have candidates containing only the same three numbers (each cell may contain 2 or 3 of them), then these three numbers must belong to these three cells. Therefore, no other cells in that unit can contain these three numbers.
Important: A triple doesn't require each cell to have exactly three candidates. For example, cells with candidates {4,9}, {1,4}, and {1,9} still form a triple because these three cells collectively use {1,4,9}.
Before reading this article, we recommend understanding Sudoku naming conventions and Naked Pairs, which will help you understand the analysis examples below.
Example 1: Naked Triples in a Row
Let's look at the first example, where we find a Naked Triple in Row 4.
Analysis Process
From the diagram, we can see the candidates for each cell in Row 4:
- R4C1 = 7 (solved)
- R4C2 = {2,4,5,9}
- R4C3 = {4,5,6}
- R4C4 = 3 (solved)
- R4C5 = {2,6}
- R4C6 = {4,9}
- R4C7 = {1,4}
- R4C8 = {1,9}
- R4C9 = 8 (solved)
- R4C2 = {2,4,5,9} contains 4 and 9, remove 4 and 9
- R4C3 = {4,5,6} contains 4, remove 4
In Row 4, R4C6{4,9}, R4C7{1,4}, and R4C8{1,9} form a Naked Triple {1,4,9}.
Action: Remove candidates 4 and 9 from R4C2, remove candidate 4 from R4C3.
Example 2: Naked Triples in a Box
Now let's look at another example, finding a Naked Triple in Box 2 (the top-center 3×3 region).
Analysis Process
From the diagram, we can see the candidates for each cell in Box 2:
- R1C4 = {2,6,7}
- R1C5 = {2,3,7}
- R1C6 = 8 (solved)
- R2C4 = {4,9}
- R2C5 = {3,4,9}
- R2C6 = 1 (solved)
- R3C4 = 5 (solved)
- R3C5 = {3,4,9}
- R3C6 = {4,6,7,9}
- R1C5 = {2,3,7} contains 3, remove 3
- R3C6 = {4,6,7,9} contains 4 and 9, remove 4 and 9
In Box 2, R2C4{4,9}, R2C5{3,4,9}, and R3C5{3,4,9} form a Naked Triple {3,4,9}.
Action: Remove candidate 3 from R1C5, remove candidates 4 and 9 from R3C6.
Variations of Naked Triples
Naked Triples have multiple variations, the key being that three cells collectively use three numbers:
| Variation Type | Candidates in Three Cells | Description |
|---|---|---|
| Complete (3-3-3) | {1,2,3}, {1,2,3}, {1,2,3} | All three cells have all three candidates |
| 2-3-3 Type | {4,9}, {3,4,9}, {3,4,9} | One cell has 2 candidates, two have 3 (Example 2) |
| 2-2-3 Type | {1,2}, {2,3}, {1,2,3} | Two cells have 2 candidates, one has 3 |
| 2-2-2 Type | {4,9}, {1,4}, {1,9} | All three cells have only 2 candidates (Example 1, hardest to spot) |
To identify a Naked Triple: combine all candidates from three cells. If the result contains exactly three different numbers, they form a Naked Triple. For example, {4,9} ∪ {1,4} ∪ {1,9} = {1,4,9}, only 3 numbers, therefore it's a Naked Triple.
Naked Pairs vs Naked Triples
Let's compare Naked Pairs and Naked Triples:
| Comparison | Naked Pairs | Naked Triples |
|---|---|---|
| Number of Cells | 2 cells | 3 cells |
| Number of Digits | 2 digits | 3 digits |
| Candidate Requirement | Both cells have identical candidates | Three cells have subsets of same three digits |
| Recognition Difficulty | Easier | Harder (more variations) |
| Elimination Effect | Eliminates 2 digits | Eliminates 3 digits |
How to Find Naked Triples?
Finding Naked Triples requires a systematic approach:
- Three cells must be in the same unit (row/column/box) to form a Naked Triple
- Can only eliminate candidates from the unit where the triple exists, cannot eliminate across units
- If three cells' combined candidates exceed 3 numbers, e.g., {1,2}, {2,3}, {3,4}, they do not form a Naked Triple (4 different numbers: 1,2,3,4)
- Easy to miss 2-2-2 type Naked Triples (when all three cells have only 2 candidates)
Technique Summary
Key points for applying Naked Triples:
- Search condition: Three cells must be in the same row, column, or box
- Candidate requirement: Combined candidates of three cells must be exactly three numbers
- Variation recognition: Each cell doesn't need three candidates; {4,9}, {1,4}, {1,9} is also a Naked Triple
- Elimination scope: Can only eliminate candidates from other cells in the same unit
- Note: Naked Triples don't directly give answers, but simplify the puzzle by eliminating candidates
Advanced: Naked Quads
Naked Triples can be extended to Naked Quads: When four cells in the same unit have candidates that are subsets of four numbers, those four numbers can be eliminated from other cells. However, quads are relatively rare and harder to identify in practice.
Start a Sudoku game and try using Naked Triples to find candidates you can eliminate!