Solving Tips
XY-Wing Technique: Elegant Elimination Using Three Bi-value Cells
XY-Wing is an elegant advanced Sudoku technique that uses the special relationship between three bi-value cells (cells with exactly two candidates) for logical elimination.
Core Principle:
An XY-Wing consists of three bi-value cells: one Pivot and two Wings. The pivot must be able to "see" both wing cells (i.e., share the same row, column, or box). If the pivot is {X,Y}, one wing is {X,Z}, and the other wing is {Y,Z}, then Z must be in one of the wing cells. Therefore, any cell that can see both wings cannot contain Z.
An XY-Wing consists of three bi-value cells: one Pivot and two Wings. The pivot must be able to "see" both wing cells (i.e., share the same row, column, or box). If the pivot is {X,Y}, one wing is {X,Z}, and the other wing is {Y,Z}, then Z must be in one of the wing cells. Therefore, any cell that can see both wings cannot contain Z.
XY-Wing Diagram: Pivot {X,Y} with Wings {X,Z} and {Y,Z} - Z must be in Wing 1 or Wing 2
Before reading this article, we recommend understanding Sudoku naming conventions and the basics of Naked Pairs.
XY-Wing Structure
An XY-Wing contains three key elements:
- Pivot: The central cell with candidates {X,Y}, must be able to see both wing cells
- Wing 1: Candidates {X,Z}, shares a row, column, or box with the pivot
- Wing 2: Candidates {Y,Z}, shares a row, column, or box with the pivot
Key feature: The three cells share three digits X, Y, Z, with each digit appearing exactly twice.
Why Does XY-Wing Work?
1
Pivot can only be X or Y: The pivot cell {X,Y} must ultimately contain either X or Y.
2
If pivot is X: Wing 1 {X,Z} cannot be X (no duplicates in the same unit), so Wing 1 must be Z.
3
If pivot is Y: Wing 2 {Y,Z} cannot be Y (no duplicates in the same unit), so Wing 2 must be Z.
4
Conclusion: Whether the pivot is X or Y, Z must be in either Wing 1 or Wing 2. Therefore, any cell that can see both wings cannot contain Z.
Example 1: XY-Wing with R7C5 as Pivot
Let's look at the first example showing a typical XY-Wing structure.
Figure 1: Pivot R7C5{6,9}, Wings R8C4{5,6} and R7C7{5,9}, eliminate 5 from R8C7
Analysis Process
1
Identify the pivot: R7C5 is a bi-value cell with candidates {6, 9}.
2
Find wing cells:
- R8C4 (Wing 1): candidates {5, 6}, shares Box 8 with pivot
- R7C7 (Wing 2): candidates {5, 9}, shares Row 7 with pivot
3
Verify XY-Wing structure:
- Pivot {6,9} + Wing 1 {5,6} + Wing 2 {5,9} = three digits 5, 6, 9 each appearing twice ✓
- Pivot can see both wings (Box 8 and Row 7) ✓
- Common digit Z = 5
4
Reasoning process:
- If R7C5=6 → R8C4 cannot be 6 → R8C4=5
- If R7C5=9 → R7C7 cannot be 9 → R7C7=5
- Either way, R8C4 or R7C7 must contain 5
5
Find elimination target: R8C7 can see both wings (same row as R8C4, same box as R7C7).
Conclusion:
XY-Wing: Pivot R7C5, Wings R8C4 and R7C7.
Eliminate candidate 5 from R8C7.
XY-Wing: Pivot R7C5, Wings R8C4 and R7C7.
Eliminate candidate 5 from R8C7.
Example 2: XY-Wing with R6C3 as Pivot
Now let's look at another example showing a different positional relationship.
Figure 2: Pivot R6C3{6,8}, Wings R1C3{6,9} and R6C7{8,9}, eliminate 9 from R1C7
Analysis Process
1
Identify the pivot: R6C3 is a bi-value cell with candidates {6, 8}.
2
Find wing cells:
- R1C3 (Wing 1): candidates {6, 9}, shares Column 3 with pivot
- R6C7 (Wing 2): candidates {8, 9}, shares Row 6 with pivot
3
Verify XY-Wing structure:
- Pivot {6,8} + Wing 1 {6,9} + Wing 2 {8,9} = three digits 6, 8, 9 each appearing twice ✓
- Pivot can see both wings (Column 3 and Row 6) ✓
- Common digit Z = 9
4
Reasoning process:
- If R6C3=6 → R1C3 cannot be 6 → R1C3=9
- If R6C3=8 → R6C7 cannot be 8 → R6C7=9
- Either way, R1C3 or R6C7 must contain 9
5
Find elimination target: R1C7 can see both wings (same row as R1C3, same column as R6C7).
Conclusion:
XY-Wing: Pivot R6C3, Wings R1C3 and R6C7.
Eliminate candidate 9 from R1C7.
XY-Wing: Pivot R6C3, Wings R1C3 and R6C7.
Eliminate candidate 9 from R1C7.
How to Find XY-Wings
Finding XY-Wings requires a systematic approach:
1
Find all bi-value cells: First, mark all cells that have exactly two candidates.
2
Select potential pivots: For each bi-value cell {X,Y}, check other bi-value cells it can see.
3
Look for matching wings: Find two bi-value cells where one contains X and a third digit Z, and the other contains Y and Z.
4
Verify structure: Confirm the pivot can see both wing cells.
5
Find elimination targets: Find cells that can see both wings and contain candidate Z.
Important Notes:
- The pivot must be able to see both wing cells (share row, column, or box)
- The two wing cells do not need to see each other
- Eliminate the common digit Z, which is the digit shared by both wings
- Elimination targets must be able to see both wings
Technique Summary
Key points for applying XY-Wing:
- Recognition: Three bi-value cells with candidates {X,Y}, {X,Z}, {Y,Z}
- Structure requirement: Pivot {X,Y} can see both wings {X,Z} and {Y,Z}
- Elimination target: The common digit Z
- Elimination scope: All cells that can see both wing cells
Practice Now:
Start a Sudoku game and try using XY-Wing for elimination! When you find multiple bi-value cells, check if they can form an XY-Wing structure.
Start a Sudoku game and try using XY-Wing for elimination! When you find multiple bi-value cells, check if they can form an XY-Wing structure.