XYZ-Wing in Sudoku: The Complete Guide

XYZ-Wing in Sudoku: The Complete Guide

What is an XYZ-Wing?

An XYZ-Wing is an advanced Sudoku technique involving three cells in a specific pattern. It's closely related to the XY-Wing but with one key difference: the pivot cell has three candidates instead of two.

The pattern consists of:

  • Pivot cell: Contains exactly three candidates {X, Y, Z}
  • Wing cell 1: Contains {X, Z} and can see the pivot
  • Wing cell 2: Contains {Y, Z} and can see the pivot

The Elimination Rule:

If any cell can see all three cells (pivot + both wings), you can eliminate candidate Z from that cell. Why? Because Z must be placed in one of these three cells, so it cannot be anywhere else that sees all three.

Example:

Pivot at Row 5, Col 5 with {2, 4, 7}. Wing 1 at Row 5, Col 2 with {2, 7}. Wing 2 at Row 2, Col 5 with {4, 7}. Any cell that sees all three cells cannot contain 7, because 7 must go in one of these three cells.

Why XYZ-Wing Matters

  • It extends XY-Wing logic to three-candidate pivots.
  • It appears more frequently than XY-Wing in some advanced puzzles.
  • It provides eliminations when simpler techniques fail.
  • It's essential for advanced puzzle solving.
  • It demonstrates sophisticated pattern recognition skills.

How to Spot an XYZ-Wing

  1. Find a pivot cell with three candidates.
    Look for cells with exactly three candidates like {1,3,5} or {2,6,8}. Call these candidates X, Y, and Z.
  2. Look for wing cells with two candidates.
    Find two cells that each have two candidates: one contains {X,Z} and the other contains {Y,Z}. Notice both share Z with the pivot.
  3. Verify visibility.
    The pivot must be able to "see" both wing cells (same row, column, or box).
  4. Find elimination targets.
    Look for cells that see all three cells (pivot and both wings).
  5. Eliminate Z.
    Remove candidate Z from any cell that sees all three cells in the pattern.

XYZ-Wing vs XY-Wing

These related techniques differ in one key aspect:

  • XY-Wing:
    • Pivot: {X, Y} (two candidates)
    • Wing 1: {X, Z}
    • Wing 2: {Y, Z}
    • Elimination: Z from cells seeing both wings (pivot doesn't need to be seen)
  • XYZ-Wing:
    • Pivot: {X, Y, Z} (three candidates, includes Z)
    • Wing 1: {X, Z}
    • Wing 2: {Y, Z}
    • Elimination: Z from cells seeing all three cells (including pivot)

Think of XYZ-Wing as "XY-Wing plus one"—the pivot gains candidate Z, and elimination targets must see the pivot too.

Step-by-Step Example

Let's identify an XYZ-Wing in a puzzle:

Pattern cells:

  • Pivot: Row 4, Col 5 contains {3, 6, 8}
  • Wing 1: Row 4, Col 2 contains {3, 8} (shares row with pivot)
  • Wing 2: Row 1, Col 5 contains {6, 8} (shares column with pivot)

Verification:

  • ✓ Pivot has three candidates: {3, 6, 8}
  • ✓ Wing 1 has {3, 8}—shares 3 and 8 with pivot
  • ✓ Wing 2 has {6, 8}—shares 6 and 8 with pivot
  • ✓ Common candidate across all three: 8
  • ✓ Pivot sees both wings (same row/column)

The Logic:

Candidate 8 must go in one of these three cells:

  • If 8 goes in the pivot → wings must be 3 and 6
  • If 8 goes in Wing 1 → pivot must have 6, Wing 2 must have 6 (contradiction if both need 6)
  • Actually, if 8 is in Wing 1, pivot uses 3 or 6, Wing 2 uses 6
  • If 8 is in Wing 2 → pivot uses 3 or 6, Wing 1 uses 3

Elimination:

Any cell that can see Row 4/Col 5 (pivot), Row 4/Col 2 (Wing 1), AND Row 1/Col 5 (Wing 2) cannot contain 8.

For example, Row 1, Col 2 (intersection of Wing 1's column and Wing 2's row) sees all three cells. If it contains candidate 8, eliminate it.

Visual Example: The Basic Pattern

  • Setup: You find three cells forming a small "Y" or "T" shape in the grid.
  • Pivot: Cell at Row 6, Col 6 in Box 5 (center) with {1, 5, 9}
  • Wing 1: Cell at Row 6, Col 3 with {1, 9} (same row)
  • Wing 2: Cell at Row 8, Col 6 with {5, 9} (same column)
  • Common candidate: All three cells contain 9
  • Elimination target: Row 8, Col 3 sees all three cells
  • Action: Remove 9 from Row 8, Col 3 if it exists

Strategies for Spotting XYZ-Wing Quickly

  1. Start with three-candidate cells
    Scan for cells with exactly three candidates. These are your potential pivots.
  2. Look for bi-value neighbors
    From each potential pivot, check cells in the same row, column, or box for cells with exactly two candidates.
  3. Check for the Z pattern
    Verify that both wing cells share one common candidate with the pivot (the Z value).
  4. Geometric awareness
    XYZ-Wings often form L-shapes or T-shapes on the grid. Train your eye to spot these configurations.
  5. Use after XY-Wing searches
    When scanning for XY-Wings, also note three-candidate cells. You're already in the right mindset.

Common Pitfalls

  • Wrong pivot: The pivot must have exactly three candidates, not two or four.
  • Missing the common Z: All three cells must share one common candidate (Z).
  • Incorrect elimination: Only eliminate Z from cells that see ALL THREE cells, not just two.
  • Visibility errors: Make sure the pivot truly sees both wing cells (same row, column, or box).
  • Confusing with XY-Wing: Remember, XYZ-Wing pivots have three candidates; XY-Wing pivots have two.

Practice: Find the XYZ-Wing

Try identifying this pattern:

Given cells:

  • Row 3, Col 7: {2, 4, 7}
  • Row 3, Col 4: {2, 7}
  • Row 5, Col 7: {4, 7}
  • Row 9, Col 4: {1, 3, 6, 8}
  • Row 5, Col 4: {1, 2, 4, 5, 7, 8}

Question: Is there an XYZ-Wing pattern? If so, what can be eliminated?

Solution:

  • Pivot: Row 3, Col 7 with {2, 4, 7}
  • Wing 1: Row 3, Col 4 with {2, 7} (same row)
  • Wing 2: Row 5, Col 7 with {4, 7} (same column)
  • Common candidate: 7 appears in all three cells
  • XYZ-Wing confirmed!

Elimination: Find cells that see all three:

  • Row 5, Col 4 sees Wing 1 (column 4), Wing 2 (row 5), and Pivot? Let's check...
  • Row 5, Col 4 is in Row 5 (sees Wing 2), Column 4 (sees Wing 1), but doesn't see Row 3, Col 7 (pivot) directly...
  • Actually, we need to check Box relationships too. Row 3, Col 7 is in Box 3. Row 5, Col 4 is in Box 5. They don't see each other.

The elimination would apply to cells in Row 3 that are also in Column 7's box, or in Row 5 that are in Column 4's box, creating a cell that sees all three through row/column/box relationships.

Why XYZ-Wing Sets the Stage

XYZ-Wing bridges intermediate and expert techniques:

  • It extends familiar XY-Wing logic to a new scenario
  • It prepares you for WXYZ-Wing (four-cell version)
  • It demonstrates how adding one candidate changes elimination rules
  • It improves your ability to track complex cell relationships
  • It's practical enough to appear regularly in advanced puzzles

Quick Recap

Technique Pivot Candidates Wing Pattern Elimination Rule Difficulty
XY-Wing 2 (X,Y) {X,Z} & {Y,Z} Sees both wings Advanced
XYZ-Wing 3 (X,Y,Z) {X,Z} & {Y,Z} Sees all 3 cells Advanced
WXYZ-Wing 4 (W,X,Y,Z) Three wings Sees all 4 cells Advanced

Final Thought

When you're stuck on an advanced puzzle, look for three-candidate cells that might be XYZ-Wing pivots. Does a three-candidate cell have two bi-value neighbors sharing a common candidate? You might have found an XYZ-Wing!

Frequently Asked Questions

What is an XYZ-Wing in Sudoku?

An XYZ-Wing is an advanced Sudoku technique involving three cells: a pivot cell with candidates {X,Y,Z} and two wing cells with candidates {X,Z} and {Y,Z}. If any cell can see all three cells of the XYZ-Wing, you can eliminate candidate Z from that cell, since Z must appear in one of the three wing cells.

How do I spot an XYZ-Wing?

To spot an XYZ-Wing: 1) Find a cell with exactly three candidates {X,Y,Z}, this is your pivot, 2) Look for two cells with two candidates each: {X,Z} and {Y,Z}, 3) Verify the pivot sees both wing cells, 4) Find cells that see all three cells in the pattern, 5) Eliminate Z from any such cells.

What's the difference between XYZ-Wing and XY-Wing?

XY-Wing has a pivot with two candidates {X,Y} and wings {X,Z} and {Y,Z}, eliminating Z from cells that see both wings. XYZ-Wing has a pivot with three candidates {X,Y,Z} and wings {X,Z} and {Y,Z}, eliminating Z from cells that see all three cells including the pivot. XYZ-Wing is slightly more restrictive but follows similar logic.

Why is XYZ-Wing important?

XYZ-Wing is important because it extends XY-Wing logic to three-candidate pivots, it appears more frequently than XY-Wing in some puzzles, it provides eliminations when simpler techniques fail, and it's essential for advanced puzzle solving.

When should I look for XYZ-Wing?

Look for XYZ-Wing after exhausting basic and intermediate techniques. Focus on cells with exactly three candidates as potential pivots. Check if you can find two bi-value cells (two candidates each) that form the wing pattern. This technique is more common in advanced puzzles.

Ready to advance? Check out our complete strategy guide for more techniques.

Related Strategies

Once you've mastered XYZ-Wing, these techniques build naturally:

  • XY-Wing - The foundation technique with two-candidate pivots
  • WXYZ-Wing - Four-cell extension of XYZ-Wing (coming soon)
  • X-Wing - Different pattern recognition approach
  • Swordfish - Advanced fish pattern
  • Naked Triples - Three-cell subset technique

Practice XYZ-Wing

Next up: Try WXYZ-Wing to extend this logic further (coming soon).

← Back to All Strategies

All Strategies