Naked Triples in Sudoku: Extending the Pair Logic
A Naked Triple is a Sudoku solving technique where three cells within a single row, column, or 3x3 block collectively contain only three candidate numbers among them. This pattern allows you to eliminate those three candidate digits from all other cells in that same unit, which often reveals [Naked Singles](/techniques/naked-singles) and enables further progress. It is a direct and powerful extension of the simpler [Naked Pair](/techniques/naked-pairs) logic, applying the same principle to a trio of cells for solving more challenging puzzles.
What Exactly is a Naked Triple?
A Naked Triple consists of three cells that belong to the same unit—a row, column, or 3x3 block. The defining condition is that the only possible candidates for these three cells, when considered as a group, are exactly three different digits. A crucial detail is that the three cells do not each need to contain all three candidates. For example, one cell might have candidates (1,2), another (2,3), and the third (1,3). As a set, they cover the digits 1, 2, and 3, forming a valid Naked Triple.
How It Extends from Naked Pair Logic
The logic of a Naked Triple is a natural progression from a Naked Pair. A Naked Pair involves two cells in a unit that share the same two candidates, locking those digits into those two spots. A Naked Triple scales this up: three cells are collectively 'reserving' three digits. The core principle remains identical: if a set of N cells contains exactly N candidate digits, those digits cannot appear anywhere else in the unit. This is a foundational Sudoku strategy known as subset or subset elimination.
How to Spot a Naked Triple
To find a Naked Triple, scan each row, column, and block. Look for three cells where the pool of possible digits is limited to only three different numbers. You are looking for combinations like (1,2), (2,3), and (1,3), or (1,2), (1,2), and (1,2,3). The key is to mentally combine the candidate notes from the three cells. If the total number of distinct digits is three, you have found a Naked Triple. This technique is most useful in the mid-game after you have filled in obvious candidates.
- Focus on units where many cells already have few candidates.
- Check for two cells that form a strong pair first; a third cell that shares some of their candidates might complete a triple.
The Elimination Rule: What to Remove
Once you confirm a Naked Triple, you can eliminate the three involved candidate digits from every other cell in that row, column, or block. For instance, if cells A, B, and C in a row form a Naked Triple for digits {4,7,9}, you can remove 4, 7, and 9 from the candidate notes of all other cells in that row. This cleanup is the primary benefit of the technique, as it simplifies the puzzle and frequently creates new Naked Singles or Hidden Pairs.
Avoiding the Common Trap
The most common misunderstanding is thinking all three cells must contain the same three candidates. This is not required. A valid Naked Triple can be formed by cells with candidate sets like (1,2), (2,3), and (3). The combined set is {1,2,3}, which is three digits across three cells. As long as no cell in the triple contains a candidate outside the set of three, and the total number of distinct digits is exactly three, the pattern holds. Recognizing this flexibility is key to applying the technique effectively.
Key Facts
- ▪A Naked Triple is three cells in a single Sudoku unit (row, column, or block) that collectively contain only three possible candidate digits.
- ▪The three cells do not need to each have all three candidates; one cell might have only one or two of them.
- ▪The logic is an extension of the Naked Pair technique, applying the same subset elimination principle to three cells.
- ▪Once identified, the three candidate digits can be eliminated from all other cells in that unit.
- ▪This elimination often directly reveals Naked Singles, where a cell has only one possible number left.
- ▪Naked Triples are a mid-difficulty technique crucial for solving Hard and Expert-level Sudoku puzzles.
- ▪Scanning for cells with few candidates (2 or 3) in a crowded unit is a good strategy for spotting potential triples.
- ▪A Naked Triple is different from a Hidden Triple, where three digits are hidden among more candidates in three cells.
Frequently Asked Questions
Do all three cells in a Naked Triple need the same three candidates?
No. The cells can have different combinations like (1,2), (2,3), and (1,3). The group must share exactly three distinct digits total.
What is the difference between a Naked Triple and a Hidden Triple?
In a Naked Triple, the three digits are the *only* candidates in those three cells. In a Hidden Triple, the three digits are hidden among other candidates in three cells, and you eliminate the extra candidates.
Can a Naked Triple exist in a 3x3 block as well as a row?
Yes. A Naked Triple can occur in any single unit: a row, a column, or a 3x3 block. The rule and eliminations apply only within that specific unit.
Is finding a Naked Triple always helpful?
Yes. It always allows for candidate elimination, which simplifies the puzzle. It is a decisive step that often unlocks the next move.