XY-Wing Sudoku Technique Explained

The gateway to forcing chains and advanced logic

XY-Wing example in Sudoku

What is an XY-Wing?

An XY-Wing (also called Y-Wing) is a pattern involving three bi-value cells that creates a logical chain. The "pivot" cell sees two "wing" cells, and together they guarantee that a specific candidate can be eliminated from cells that see both wings.

It's called XY-Wing because the three cells contain candidates X, Y, and Z in a specific arrangement: {XY}, {XZ}, and {YZ}.

The XY-Wing Principle: If a pivot cell {XY} sees wing cells {XZ} and {YZ}, then any cell that sees BOTH wings cannot contain Z.

The Three Cells

Pivot Cell

Contains candidates {X, Y}

Sees both wing cells

The "hinge" of the pattern

Wing 1

Contains candidates {X, Z}

Sees the pivot cell

Shares X with pivot

Wing 2

Contains candidates {Y, Z}

Sees the pivot cell

Shares Y with pivot

Why XY-Wing Works

The Forcing Logic

The pivot cell must be either X or Y. Consider both cases:

  1. If pivot = X: Wing 1 {XZ} cannot be X, so it must be Z
  2. If pivot = Y: Wing 2 {YZ} cannot be Y, so it must be Z

Conclusion: Either Wing 1 or Wing 2 must be Z. Any cell that sees BOTH wings cannot be Z!

How to Find XY-Wings

Step-by-Step Process:

  1. Find bi-value cells: Identify all cells with exactly two candidates
  2. Choose a potential pivot: Pick a bi-value cell {XY}
  3. Find Wing 1: Look for a bi-value cell that sees the pivot and shares one candidate (X), with the form {XZ}
  4. Find Wing 2: Look for a bi-value cell that sees the pivot and shares the other candidate (Y), with the form {YZ}
  5. Verify Z is common: Both wings must share the same "Z" candidate
  6. Find elimination targets: Look for cells containing Z that see BOTH wings
  7. Eliminate Z: Remove Z from those target cells

Example Analysis

Finding an XY-Wing

  • Pivot (R2C5): {3, 7}
  • Wing 1 (R2C8): {3, 9} – shares 3 with pivot, in same row
  • Wing 2 (R5C5): {7, 9} – shares 7 with pivot, in same column

XY-Wing found! Z = 9 (the common candidate in both wings)

Elimination: Remove 9 from any cell that sees both R2C8 AND R5C5.

Finding Elimination Targets

A cell can "see" another cell if they share a row, column, or box. For XY-Wing eliminations, the target cell must see BOTH wings simultaneously.

Common target locations:

  • Cells in the same box as both wings
  • Cells at the intersection of the wings' row and column
  • Any cell that shares a unit with each wing

Common Mistakes

  • Wrong pivot: The pivot must see BOTH wings directly
  • Mismatched Z: Both wings must share the same third candidate
  • Wrong eliminations: Only eliminate from cells that see BOTH wings, not just one
  • Non-bi-value cells: All three cells must have exactly two candidates
Pro Tip: XY-Wings are easier to find when you have many bi-value cells. After applying simpler techniques, scan your bi-value cells systematically. For each potential pivot, check if suitable wings exist.

XY-Wing Variations

The XY-Wing concept extends to more complex patterns:

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