XYZ-Wing Sudoku Explained: Advanced Elimination Technique

The XYZ-Wing technique is an advanced Sudoku solving method that extends the XY-Wing concept by incorporating a trivalue cell. This powerful elimination technique can solve puzzles that other methods cannot crack.

What is XYZ-Wing?

XYZ-Wing is a three-cell pattern that uses one trivalue cell and two bivalue cells to eliminate candidates. The name comes from the three candidates involved: X, Y, and Z.

XYZ-Wing Structure

An XYZ-Wing consists of:

• Pivot Cell (XYZ): Contains three candidates X, Y, and Z
• Pincer Cell 1 (XZ): Contains candidates X and Z, shares a unit with pivot
• Pincer Cell 2 (YZ): Contains candidates Y and Z, shares a unit with pivot

The Logic Behind XYZ-Wing

The elimination logic works as follows:

1. If the pivot cell contains X, then Pincer Cell 1 must contain Z
2. If the pivot cell contains Y, then Pincer Cell 2 must contain Z
3. If the pivot cell contains Z, then both pincer cells cannot contain Z
4. In all cases, Z cannot be in cells that see both pincer cells

How to Identify XYZ-Wing Patterns

Step 1: Find Trivalue Cells

Look for cells that contain exactly three candidates. These are your potential pivot cells.

Step 2: Locate Bivalue Pincer Cells

For each trivalue cell, look for two bivalue cells that:

• Share a unit with the pivot
• Each contains two of the pivot's three candidates
• Both contain the same third candidate

Step 3: Verify the Pattern

Ensure that the two pincer cells share a common candidate that's different from the pivot's candidates.

Step 4: Apply the Elimination

Find cells that can see both pincer cells and eliminate the shared candidate from those cells.

XYZ-Wing Examples

Example 1: Basic XYZ-Wing

Consider a pivot cell with (1,2,3), a pincer cell with (1,3), and another pincer cell with (2,3). Any cell seeing both pincer cells cannot contain 3.

Example 2: Complex XYZ-Wing

XYZ-Wing patterns can span multiple boxes and involve complex unit relationships.

XYZ-Wing vs XY-Wing

Key differences between XYZ-Wing and XY-Wing:

• XYZ-Wing: Uses one trivalue cell as pivot
• XY-Wing: Uses bivalue cells only
• XYZ-Wing: More powerful but harder to spot
• XY-Wing: More common in typical puzzles

Advanced XYZ-Wing Applications

Extended XYZ-Wing

In some cases, the XYZ-Wing pattern can be extended to involve more cells while maintaining the same logical structure.

Multiple XYZ-Wings

Complex puzzles may contain multiple XYZ-Wing patterns that can be applied sequentially.

Common Mistakes to Avoid

Mistake 1: Wrong Cell Types

Remember that XYZ-Wing requires one trivalue cell and two bivalue cells.

Mistake 2: Incorrect Unit Sharing

Ensure that pincer cells actually share a unit with the pivot cell.

Mistake 3: Wrong Elimination Target

Only eliminate the shared candidate from cells that can see both pincer cells.

When to Use XYZ-Wing

XYZ-Wing is most effective when:

• Other techniques have been exhausted
• The puzzle contains trivalue cells
• You're looking for advanced elimination methods
• Standard XY-Wing patterns aren't available

Practice Tips

• Look for trivalue cells first
• Check for bivalue cells that share units
• Verify the candidate relationships
• Practice with advanced puzzle collections

Related Techniques

XYZ-Wing is part of the Wing family of techniques:

• XY-Wing - Bivalue cell version
• Y-Wing - Similar three-cell pattern
• XYZ-Wing Basics - Fundamental concepts