Wings
XY-Wing
An XY-Wing is three cells holding three digits between them. A pivot cell has two candidates; two wing cells each share one of them with the pivot and carry a third digit in common. Whichever way the pivot resolves, one of the two wings must take that shared third digit, so it can be removed from any cell that can see both wings at once.
- Level
- Advanced
The pattern
What it does.
Name the pivot {X, Y} and the wings {X, Z} and {Y, Z}. If the pivot is X, the first wing is forced to Z; if the pivot is Y, the second wing is. Either way a Z is placed in one of the two wings.
Because a Z is certain to land in one wing or the other, no cell that can see both wings can be a Z. That's the elimination, and it's the only one the pattern gives.
The wings don't need to see each other, and they don't need to be in the same unit as one another. Each only has to see the pivot. That's what makes the pattern common and the search fiddly.
Example
A worked example.
Find a pivot
Look for a cell with exactly two candidates. Any such cell can be the pivot of an XY-Wing, so the technique starts from the pairs you already have.
Look for two wings in its units
Search the cells that share a row, a column or a box with the pivot for another two-candidate cell that shares one of the pivot's digits. You need two of them, and they must not share both.
Check the third digit matches
The two wings must both carry the same third digit, the one the pivot doesn't have. If they carry different third digits, this isn't an XY-Wing.
Clear the third digit
Remove the shared third digit from every cell that can see both wings at once. Those cells are usually the ones sitting between them, and often none exist, in which case the pattern is real but useless.
Spotting it
How to spot it.
- Start from two-candidate cells, not from a digit. An XY-Wing is a shape made of pairs, so a grid with plenty of pencil marks is where it turns up.
- A pivot usually has its two wings in different units: one in its row, one in its box, or one in its column. Look in all three.
- Check that the wings have somewhere to eliminate before working through the logic. A wing pair with no common observer costs more to find than it gives back.
- If the third digit is the same in both wings and the wings themselves see each other, you have something stronger than an XY-Wing and should read it as one.
Where it stops working
An XY-Wing removes one digit and places nothing, so it's easy to prove and easy to waste time on. Every cell involved must have exactly two candidates: a third candidate in the pivot or in either wing turns it into a different pattern and the elimination becomes wrong. It also depends on complete pencil marks, more than any technique before it, because the whole shape is built out of them.
Keep going
Related topics.
New to sudoku? The rules take two minutes to read, and the first-puzzle walkthrough covers the basics that come before any of these.
Beyond the lessons.
The game's lessons stop at X-Wing, so there's no XY-Wing lesson. The eight it does teach will get you through almost any puzzle; this page is here for when you run into the pattern anyway.