XYZ-Wing

XYZ-Wing builds on the XY-Wing pattern by allowing the pivot cell to contain all three candidates. This small change expands where the technique applies while requiring a stricter elimination condition.

What is an XYZ-Wing?

An XYZ-Wing consists of three cells:

• Pivot cell: Contains candidates [X,Y,Z] — three candidates!
• Wing 1: Contains [X,Z], can "see" the pivot
• Wing 2: Contains [Y,Z], can "see" the pivot

The key difference from XY-Wing: the pivot has Z too.

Visual Pattern

The three cells:

• Pivot A1: [X,Y,Z] — the "hinge" with three candidates
• Wing 1 at C1: [X,Z] — shares X and Z with pivot
• Wing 2 at A3: [Y,Z] — shares Y and Z with pivot

The Logic Explained

Think about what the pivot could become:

If the pivot becomes X:

1. Wing 1 can't be X (they see each other)
2. Wing 1 must be Z
3. Z is placed!

If the pivot becomes Y:

1. Wing 2 can't be Y (they see each other)
2. Wing 2 must be Z
3. Z is placed!

If the pivot becomes Z:

1. Z is placed directly in the pivot!

In all three cases, Z appears somewhere among these three cells!

The Elimination Rule

Here's the critical difference from XY-Wing:

For XYZ-Wing, because the pivot could be Z, elimination targets must see the pivot too — not just the wings.

Finding elimination targets:

1. Identify the pivot and both wings
2. Find cells that can "see" ALL THREE cells
3. Eliminate Z from those cells

This is more restrictive than XY-Wing, so eliminations are often limited to cells in the same box as the pivot.

Concrete Example

The XYZ-Wing:

• X = 7, Y = 3, Z = 9
• Pivot at A: [3,7,9]
• Wing 1 at D: [7,9] — shares 7,9 with pivot (same column)
• Wing 2 at H: [3,9] — shares 3,9 with pivot (same box)

Elimination: Cell B is in the same box as the pivot and same row as Wing 1. It sees all three cells. Eliminate 9 from B!

Comparing XY-Wing and XYZ-Wing

How to Find XYZ-Wings

Method 1: Start from Triple Cells

1. Find cells with exactly 3 candidates — these are potential pivots
2. For a cell with [X,Y,Z], look for:
   • A bi-value cell [X,Z] that sees it
   • A bi-value cell [Y,Z] that sees it
3. Find cells that see all three for eliminations

Method 2: Failed XY-Wing Check

Sometimes you find an almost-XY-Wing where:

• You have three cells with the right candidate relationships
• But the "pivot" has 3 candidates instead of 2

Don't discard it — check if it's a valid XYZ-Wing!

Method 3: Box Focus

Since eliminations must see all three cells, they're often in the pivot's box:

1. Pick a box with a triple-candidate cell
2. Look for bi-value cells that could be wings
3. Check for Z candidates in cells seeing all three

The "Seeing" Requirement

For an XYZ-Wing to work:

This last requirement is what limits XYZ-Wing eliminations.

Common Configurations

Wings in Same Box

When both wings are in the pivot's box:

This is the most common XYZ-Wing configuration because finding cells that see all three is easiest when they share a box.

Wing Outside Box

Here, Wing 1 is in the same row as the pivot. Cell B1 still sees all three cells (same row as pivot and Wing 1, same column as Wing 2's column extends there).

Practice Exercises

Exercise 1: Identify the XYZ-Wing

Exercise 2: Find the elimination target

WXYZ-Wing: The Next Step

Just as XYZ-Wing extends XY-Wing, there's a WXYZ-Wing that extends further:

• Pivot: [W,X,Y,Z] — four candidates
• Wings: Bi-value cells sharing candidates with the pivot

The logic follows the same pattern, but finding valid eliminations becomes increasingly difficult as more cells must be "seen."

Common Mistakes

Mistake 1: Using XY-Wing elimination rules

XYZ-Wing requires targets to see ALL THREE cells, not just both wings. The pivot matters for eliminations here.

Mistake 2: Forgetting Z is in the pivot

The whole point of XYZ-Wing is that Z appears in all three cells. If your pivot only has two candidates, it's an XY-Wing, not XYZ-Wing.

Mistake 3: Missing valid targets

Because the requirement is strict (see all three), valid targets are often only in the pivot's box. Don't overlook them!

Mistake 4: Wrong candidate assignment

Make sure:

• Z is in all three cells
• X is shared by pivot and Wing 1
• Y is shared by pivot and Wing 2

When XYZ-Wings Appear

• Easy puzzles: Never needed
• Medium puzzles: Never needed
• Hard puzzles: Occasionally useful
• Expert puzzles: Common
• Extreme puzzles: Frequently required

XYZ-Wing is slightly less common than XY-Wing because:

1. It requires a triple-candidate pivot (less common than bi-value)
2. Eliminations are more restricted

Quick Reference

XYZ-Wing structure:

• Pivot: [X,Y,Z] — three candidates
• Wing 1: [X,Z] — sees the pivot
• Wing 2: [Y,Z] — sees the pivot
• Z: common to ALL three cells

Finding XYZ-Wings:

1. Find triple-candidate cells
2. Look for bi-value wings with shared candidates
3. Verify the X,Y,Z pattern
4. Find elimination targets seeing all three

Elimination rule:

• Eliminate Z from cells seeing ALL THREE cells
• This usually means same box as pivot
• More restrictive than XY-Wing

Why it works:

• If pivot = X → Wing 1 = Z
• If pivot = Y → Wing 2 = Z
• If pivot = Z → Pivot = Z
• Either way, Z is in one of the three cells!

What's Next?

Once you master XYZ-Wing:

• W-Wing — Four-cell pattern using strong links
• Remote Pairs — Chains of bi-value cells
• Simple Coloring — Color-based elimination technique