The XYZ-Wing Technique in Sudoku
What Is XYZ-Wing?
XYZ-Wing is a wing pattern built from one pivot cell and two wing cells. The pivot contains three candidates, X, Y, and Z. One wing contains X and Z, while the other wing contains Y and Z. The pivot must see both wings, and the two wings must share candidate Z.
Unlike XY-Wing, the pivot itself also contains Z. This changes the elimination area: a candidate Z can be removed only from cells that see the pivot and both wings. Those cells cannot contain Z because the pivot and the wings guarantee that one of the three connected cells will take Z.
How to Identify XYZ-Wing
- Find a three-candidate cell that can serve as the pivot: XYZ.
- Find a bivalue wing containing X and Z that sees the pivot.
- Find a second bivalue wing containing Y and Z that also sees the pivot.
- Confirm that the two wings share Z.
- Remove Z only from cells that can see the pivot and both wings.
The pivot and wings may be connected through rows, columns, or blocks. The two wings do not necessarily see each other. Check the visibility of each target cell carefully before making an elimination.
Why the Elimination Is Valid
Assume the pivot contains candidates 2, 5, and 8. One wing contains 2 and 8, and the other contains 5 and 8. If the pivot becomes 2, the first wing cannot be 2 and must be 8. If the pivot becomes 5, the second wing must be 8. If the pivot becomes 8, the pivot itself already contains 8.
In every possible value of the pivot, at least one of the pivot or wings must contain 8. A target cell that sees all three cannot contain 8, so candidate 8 can be eliminated there.
XYZ-Wing and XY-Wing
XY-Wing uses a two-candidate pivot XY and wings XZ and YZ. Its target cells need to see both wings. XYZ-Wing uses a three-candidate pivot XYZ, so the pivot is also part of the visibility requirement. The two techniques are related, but their elimination zones are not interchangeable.
Common Mistakes With XYZ-Wing
- The pivot must contain three candidates; a two-candidate pivot is an XY-Wing.
- Both wings must contain the same shared candidate Z.
- Do not remove Z from a cell that cannot see the pivot.
- The target cell must see the pivot and both wings, not merely two of the three cells.
When Is XYZ-Wing Useful?
XYZ-Wing is most useful after candidate notes have been updated and several bivalue cells are visible. It can remove a candidate from a carefully defined intersection even when a simpler pair or fish pattern is not available. Verify the three visibility relationships before committing the elimination.
What Should I Study Alongside XYZ-Wing?
Review XY-Wing to compare the two pivot structures, then study W-Wing for a related chain using strong links. Skyscraper offers a different way to reason about two endpoints and one candidate.