Naked Triples

What is Naked Triples?

A Naked Triple is a natural extension of the Naked Pairs concept. It occurs when three cells in the same row, column, or 3×3 box collectively contain only three distinct candidates, distributed in any combination across the three cells. The key constraint is that the union of all candidates across the three cells contains exactly three digits. Importantly, the individual cells do not each need to have all three candidates — one cell might show [1, 2], another [2, 3], and the third [1, 3]; their union {1, 2, 3} contains exactly three digits, forming a valid Naked Triple. Alternatively, one cell might show [1, 2, 3] with the other two each showing a subset. What matters is that no candidate outside the three digits appears in any of the three cells. Because the three digits {1, 2, 3} must be distributed among the three cells (each cell gets exactly one of the three), those three digits are "used up" and can be safely removed from all other cells in the unit. Naked Triples are harder to spot than Naked Pairs due to the variety of candidate distributions and the need to consider three cells at once, but they are crucial for breaking through hard puzzle stalemates.

When to Use Naked Triples

Use Naked Triples when Naked Pairs and all intersection techniques (Pointing Pairs, Box-Line Reduction) are exhausted and you need further candidate eliminations. They typically appear in Hard and Expert puzzles. To find them efficiently, focus on units that have many placed digits — fewer empty cells means fewer cells to combine. A practical approach is to start by listing all cells in a unit that have 2 or 3 candidates, then look for any group of three such cells whose combined candidate pool contains exactly 3 digits. Verification is important: double-check that no candidate in any of the three cells falls outside the three-digit pool before making eliminations.

How to Apply Naked Triples — Step by Step

1. 1

   Build or refresh the candidate grid

   Ensure your candidate grid is fully up to date. Naked Triples require accurate pencil marks across the entire board — any outdated candidate information will lead to incorrect pattern detection. If you've recently made placements, update all affected cells before searching for triples.

2. 2

   List cells with 2 or 3 candidates in a unit

   In a target row, column, or box, identify every empty cell that has exactly 2 or 3 candidates. These are the only cells that can form a Naked Triple. Cells with 4 or more candidates cannot be part of a naked triple (though they might be part of a naked quad). Write these cells and their candidates down for easy comparison.

3. 3

   Find three cells whose union is exactly 3 digits

   Test combinations of three cells from your list. For each combination, take the union of all candidates across the three cells. If the union contains exactly 3 distinct digits and nothing more, you have a Naked Triple. The configuration of which cell has which candidates is flexible: [1,2]+[2,3]+[1,3], [1,2,3]+[1,2]+[2,3], or even [1,2,3]+[1,2,3]+[1,2,3] are all valid triple forms.

4. 4

   Verify and apply elimination

   Before eliminating, confirm that all three cells are in the same unit (row, column, or box). Then remove all three digits from the candidates of every other cell in that unit. This is the payoff: after elimination, scan immediately for new Naked Singles, Hidden Singles, or Naked Pairs that have been unlocked by the removal.

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