YW Sudoku Analysis

The Geometric Logic of Bivalue Triads: A Deep Dive into the Y-Wing Strategy

1. Introduction: The "Bent" Triple

In the hierarchy of Sudoku solving techniques, the Y-Wing (formally known as the XY-Wing) represents the first major step away from single-house pattern recognition into multi-house geometry. While intermediate solvers are comfortable with "Naked Triples" (three cells in a single row, column, or box sharing three candidates), the Y-Wing applies a similar logical pressure across intersecting houses.

Fundamentally, the Y-Wing is a specific configuration of three bivalue cells (cells containing exactly two candidates). It serves as a "forcing pattern" involving exactly three distinct digits. Because of its reliance on bivalue cells and "If-Then" logic, it is mathematically classified as the shortest possible XY-Chain (Length 3).

2. Anatomy of a Y-Wing

To identify a Y-Wing, the solver must locate a specific triad of cells containing candidates A, B, and C.

2.1 The Components

  • 1. The Pivot {A, B}
    The central "Hinge" cell. It must see both Pincers.
  • 2. Pincer 1 {A, C}
    Shares house with Pivot. Shares candidate 'A'.
  • 3. Pincer 2 {B, C}
    Shares different house with Pivot. Shares candidate 'B'.

2.2 Visualization: The "Bent" Shape

Diagram 1: The Classic Layout (Candidate C = 3)
. {1,2}
Pivot		 ... ... {1,3}
Pincer 1
... ... ... ... Target?
See Both
... {2,3}
Pincer 2		 ... ... .

3. The Logical Proof

The Y-Wing works on a principle of Inescapable Truth. We do not know which Pincer contains the common candidate C, but we can prove that one of them must.

3.1 The "If-Then" Fork

If Pivot is A...

Then Pincer 1 ({A,C}) cannot be A.
Therefore, Pincer 1 is C.

If Pivot is B...

Then Pincer 2 ({B,C}) cannot be B.
Therefore, Pincer 2 is C.

Conclusion: Candidate C exists in Pincer 1 OR Pincer 2.

4. The Elimination Logic (The Kill Zone)

Because we have proven that C exists in at least one of the Pincers, we can eliminate C from any cell that intersects both Pincers.

The Rule of Intersection

The Target Cell must share a house with Pincer 1 AND a house with Pincer 2. It does not need to see the Pivot.

5. Visual Configurations

5.1 The Right-Angle Y-Wing

Forms a rectangle. Pivot is at one corner, Pincers at adjacent corners. Elimination is at the 4th corner.

5.2 The "Obtuse" (Box) Y-Wing

Pivot and Pincer 1 are in the same Box. Pincer 2 is outside but aligns with Pivot.

Diagram 2: Box Y-Wing Example
{1,2}
Pivot		 ... {2,3}
Pincer 2
{1,3}
Pincer 1		 ... X
Kill Zone

7. Common Pitfalls

  • Pivot Blindness: The Pivot must see both Pincers. If the middle cell doesn't see the others, it breaks.
  • Wrong Elimination: You can only eliminate the candidate shared by the Pincers (Candidate C). Do not eliminate A or B.
  • Invalid Candidates: Ensure the Pivot contains {A, B} while Pincers contain {A, C} and {B, C}.