Solving Tips
WXYZ-Wing Technique: Four-Cell Chain Candidate Elimination
WXYZ-Wing is a further extension of XYZ-Wing. WXYZ-Wing uses four cells that form a chain structure through shared candidates for candidate elimination. The four cells' candidates together contain exactly four different digits W, X, Y, Z.
Core Principle:
WXYZ-Wing consists of four cells that share candidate Z and form a chain relationship. A typical structure is: Pivot{W,Z}, Wing1{W,X,Z}, Wing2{X,Y,Z}, Wing3{Y,Z}. Regardless of which cell is ultimately Z, Z must be in one of these four cells. Therefore, any position that can see all four cells can have candidate Z eliminated.
WXYZ-Wing consists of four cells that share candidate Z and form a chain relationship. A typical structure is: Pivot{W,Z}, Wing1{W,X,Z}, Wing2{X,Y,Z}, Wing3{Y,Z}. Regardless of which cell is ultimately Z, Z must be in one of these four cells. Therefore, any position that can see all four cells can have candidate Z eliminated.
WXYZ-Wing diagram: Four cells form a chain relationship through shared candidates, Z must be in one of them
Before reading this article, it's recommended to understand XY-Wing and XYZ-Wing concepts first, as WXYZ-Wing is their natural extension.
Wing Technique Comparison
The evolution of Wing techniques:
| Technique | Number of Cells | Number of Candidates | Structure |
|---|---|---|---|
| XY-Wing | 3 cells | 3 digits | Pivot{X,Y} + two double-value wings |
| XYZ-Wing | 3 cells | 3 digits | Pivot{X,Y,Z} + two double-value wings |
| WXYZ-Wing | 4 cells | 4 digits | Four-cell chain structure |
Structure of WXYZ-Wing
WXYZ-Wing has multiple possible structural forms. The core requirements are:
- Four cells whose candidates together contain exactly four different digits (W, X, Y, Z)
- All four cells contain the common candidate Z
- The four cells form a chain relationship by sharing other candidates
- The four cells must be in the same unit (row, column, or box) or can be seen simultaneously by some cell
Common WXYZ-Wing structures:
1
Type 1 (2-3-3-2): Pivot{W,Z}, Wing1{W,X,Z}, Wing2{X,Y,Z}, Wing3{Y,Z}
2
Type 2 (2-2-3-3): Pivot{W,Z}, Wing1{W,X}, Wing2{X,Y,Z}, Wing3{Y,Z} (Wing1 doesn't contain Z but connects through chain)
3
Type 3 (2-2-2-4): One four-candidate cell combined with three double-candidate cells
Why Does WXYZ-Wing Work?
Taking Type 1 structure as an example:
1
Four cells share Z: Pivot{W,Z}, Wing1{W,X,Z}, Wing2{X,Y,Z}, Wing3{Y,Z} all contain candidate Z.
2
If pivot is W: Wing1{W,X,Z} cannot be W → Wing1 is X or Z. If Wing1 is X, then Wing2{X,Y,Z} cannot be X → Wing2 is Y or Z... and so on, Z must end up in some cell.
3
If pivot is Z: The pivot itself is Z.
4
Conclusion: No matter the reasoning, Z must be in one of these four cells. Therefore, positions that can see all four cells cannot have Z.
Example 1: WXYZ-Wing in a Box
Let's look at the first example showing a typical WXYZ-Wing structure.
Figure 1: WXYZ-Wing - Pivot R5C1{1,7}, Wings R6C3{1,6}, R6C4{2,6,7}, R6C7{2,6}, eliminate candidate 7 from R5C4, R5C5
Analysis Process
1
Identify WXYZ-Wing Structure:
- R5C1: candidates {1, 7}
- R6C3: candidates {1, 6}
- R6C4: candidates {2, 6, 7}
- R6C7: candidates {2, 6}
2
Verify Candidates:
- Combined candidates: {1,7} ∪ {1,6} ∪ {2,6,7} ∪ {2,6} = {1,2,6,7}
- Exactly 4 different digits (W=1, X=6, Y=2, Z=7) ✓
- Common candidate Z = 7 (appears in R5C1 and R6C4)
3
Verify Chain Relationship:
- R5C1{1,7} and R6C3{1,6} share 1
- R6C3{1,6} and R6C4{2,6,7} share 6
- R6C4{2,6,7} and R6C7{2,6} share 2 and 6
- Complete chain structure formed ✓
4
Find Elimination Targets: R5C4 and R5C5 can see all four WXYZ cells (same box or same row).
Conclusion:
WXYZ-Wing: Pivot R5C1({1,7}), Wings R6C3({1,6}), R6C4({2,6,7}), R6C7({2,6}).
Eliminate candidate 7 from R5C4, R5C5.
WXYZ-Wing: Pivot R5C1({1,7}), Wings R6C3({1,6}), R6C4({2,6,7}), R6C7({2,6}).
Eliminate candidate 7 from R5C4, R5C5.
Example 2: Cross-Unit WXYZ-Wing
Now let's look at another example showing WXYZ-Wing across different units.
Figure 2: WXYZ-Wing - Pivot R8C9{1,2}, Wings R7C3{2,5}, R7C6{4,5}, R7C8{1,4}, eliminate candidate 2 from R7C7
Analysis Process
1
Identify WXYZ-Wing Structure:
- R8C9: candidates {1, 2}
- R7C3: candidates {2, 5}
- R7C6: candidates {4, 5}
- R7C8: candidates {1, 4}
2
Verify Candidates:
- Combined candidates: {1,2} ∪ {2,5} ∪ {4,5} ∪ {1,4} = {1,2,4,5}
- Exactly 4 different digits (W=1, X=5, Y=4, Z=2) ✓
- Common candidate Z = 2 (through chain reasoning)
3
Verify Chain Relationship:
- R8C9{1,2} and R7C8{1,4} share 1
- R7C8{1,4} and R7C6{4,5} share 4
- R7C6{4,5} and R7C3{2,5} share 5
- Complete chain structure formed ✓
4
Find Elimination Target: R7C7 can see all four WXYZ cells.
Conclusion:
WXYZ-Wing: Pivot R8C9({1,2}), Wings R7C3({2,5}), R7C6({4,5}), R7C8({1,4}).
Eliminate candidate 2 from R7C7.
WXYZ-Wing: Pivot R8C9({1,2}), Wings R7C3({2,5}), R7C6({4,5}), R7C8({1,4}).
Eliminate candidate 2 from R7C7.
How to Find WXYZ-Wing?
WXYZ-Wing is more complex than XYZ-Wing and requires a more systematic approach:
1
Look for Candidate Cells: Find 4 cells in the same unit (box/row/column) whose candidates together contain exactly 4 different digits.
2
Verify Common Candidate: Confirm there's a candidate Z that appears in multiple cells (not necessarily all four, but must be provable through chain reasoning that Z must be in one of them).
3
Verify Chain Structure: The four cells must form a chain relationship by sharing candidates to ensure complete reasoning.
4
Find Elimination Targets: Find cells that can see all four cells and contain candidate Z.
Important Notes:
- The four cells' candidates must be exactly 4 different digits
- Must verify chain relationship completeness
- Elimination target must see all four cells simultaneously
- WXYZ-Wing elimination scope is usually quite limited since seeing 4 cells is required
- Recommend using Sudoku calculator for assistance as manual detection is difficult
Technique Summary
Key points for applying WXYZ-Wing:
- Identification: Four cells with candidates containing exactly 4 different digits (W, X, Y, Z)
- Structure requirement: Four cells form a chain relationship through shared candidates
- Elimination target: Common digit Z (must be in one of the four)
- Elimination range: Positions that can see all four cells
Related Techniques:
WXYZ-Wing is an advanced Wing technique. Recommended learning order:
XY-Wing → XYZ-Wing → WXYZ-Wing
After mastering these techniques, you'll be able to handle most advanced Sudoku puzzles.
WXYZ-Wing is an advanced Wing technique. Recommended learning order:
XY-Wing → XYZ-Wing → WXYZ-Wing
After mastering these techniques, you'll be able to handle most advanced Sudoku puzzles.
Practice Now:
Start a Sudoku game and try using WXYZ-Wing! Since manual detection is difficult, try using the calculator's hint feature first to familiarize yourself with this pattern.
Start a Sudoku game and try using WXYZ-Wing! Since manual detection is difficult, try using the calculator's hint feature first to familiarize yourself with this pattern.