Bent Naked Triples: An Advanced Subset Pattern
A Bent Naked Triple has two cells forming a Naked Pair in one unit (say a row) and a third cell in an intersecting unit (say a box) that holds the third candidate; eliminations apply ONLY to the unit containing the pair, not both units. This is an extension of standard [Naked Pairs](/techniques/naked-pairs) and Triples, where the pattern is 'bent' across two units, which restricts where you can make eliminations. It's a powerful pattern for clearing candidates in puzzles where simpler techniques have stalled.
Definition and Structure
A Bent Naked Triple is defined by three cells and three candidate digits, but the cells are not all in the same unit. Two of the cells must form a true Naked Pair within a single row, column, or box. This means those two cells contain only the same two candidates, like {A,B}. The third cell, which contains the third candidate {C}, must lie in a different unit that intersects the first. For example, the pair might be in a row, and the third cell is in a box that that row passes through, but not in the same row. The three digits A, B, and C form the 'triple.' The key rule: you can only eliminate candidates A and B from other cells in the unit that contains the pair. You cannot eliminate candidate C from the intersecting unit unless it forms its own separate pattern.
Visualizing the Pattern
A simple text diagram helps illustrate the 'L-shaped' formation. Consider this example within a Sudoku grid, focusing on Row 1 and Box 1 (the top-left 3x3 box).
R1C1: Candidates {A,B} R1C5: Candidates {A,B} <- These two form a Naked Pair in Row 1. R3C1: Candidate {C} <- This cell is in Box 1, but NOT in Row 1.
In this setup, cells R1C1 and R1C5 are the pair in Row 1. Cell R3C1 is the third cell, sharing Box 1 with R1C1 but not the row. The triple digits are A, B, and C. Valid eliminations: Remove candidates A and B from all other cells in Row 1 (except R1C1 and R1C5). You do NOT eliminate candidate C from other cells in Box 1 based on this pattern alone.
Common Misconceptions and Pitfalls
A major error is trying to eliminate candidates from both intersecting units. The eliminations are confined to the unit housing the Naked Pair. If the pair is in a row, you only clean up that row. You cannot eliminate the third candidate (C) from the box unless another technique, like a Hidden Pairs, applies there.
This pattern is distinct from a standard Naked Triple, where all three cells and all three candidates are contained within one unit (row, column, or box). Calling a standard triple 'bent' is incorrect. Also, the presence of this pattern does not forbid digits A, B, or C from appearing elsewhere in the intersecting box; it only provides information about the specific row containing the pair.
When and How to Look for Bent Naked Triples
This is an advanced pattern, so search for it after you have exhausted basic techniques like Naked Singles, Naked Pairs, and standard Naked Triples. Scan the grid for any strong Naked Pair. Once you spot one, check the intersecting box (or column/row) for a cell that contains a third candidate, potentially completing a triple. The third cell does not have to be 'clean'; it can contain extra candidates besides C. The logic still holds as long as the two-pair cells are clean (only A and B). Focus your scan on the areas immediately around a confirmed pair to find this efficient pattern.
Key Facts
- ▪A Bent Naked Triple consists of three cells sharing three candidate digits, but the cells are not all in the same row, column, or box.
- ▪Two of the three cells must form a true Naked Pair, containing only the same two candidates, within a single unit.
- ▪The third cell must lie in a different unit that intersects the unit containing the pair, and it must contain the third candidate of the triple.
- ▪Candidate eliminations apply only to the unit that contains the Naked Pair, not to the intersecting unit.
- ▪You eliminate the two pair candidates from all other cells in the pair's unit, which can directly lead to new Naked or Hidden Singles.
- ▪The third cell in the pattern can contain other candidates besides the triple's third digit; the pattern's logic is not invalidated.
- ▪This pattern is a logical extension of standard subset techniques and is most useful in mid-to-late puzzle solving.
- ▪A common mistake is to incorrectly eliminate candidates in the intersecting unit, which is not allowed by the pattern's rules.
Frequently Asked Questions
How is a Bent Naked Triple different from a regular Naked Triple?
A regular Naked Triple has all three cells in one unit. A Bent Triple's cells are in two intersecting units, like an L-shape. Eliminations are only valid in the unit containing the pair.
Can the third cell contain more than just the third candidate?
Yes. The third cell can have extra candidates. The critical requirement is that the two cells forming the pair contain only the two shared candidates.
Why can't I eliminate candidates from the intersecting unit?
The three cells are not a complete subset within that unit. Only the pair cells restrict the two candidates in their unit. The third cell's candidate isn't restricted in the intersecting unit by this pattern.
When should I start looking for this pattern?
Search for Bent Naked Triples after basic techniques like Naked Pairs and standard Naked Triples no longer provide progress in a puzzle.