XYZ-Wing in Sudoku

Specialist S6 Wings

An XYZ-Wing is a Y-Wing whose pivot has three candidates instead of two. The two wings have two candidates each, both drawn from the pivot's three, and they share exactly one candidate with each other. That shared candidate must occupy one of the three cells - the pivot included - so it can be removed from any cell seeing all three.

How to spot it

  • Look for a cell with exactly three candidates. That is the pivot.
  • Find two cells with exactly two candidates that the pivot sees, in a row, column or box.
  • Both wings' candidate pairs must come from the pivot's three candidates.
  • The two wings must share exactly one candidate. That is the digit you eliminate.
  • If all three cells sit in one row, column or box, it is a naked triple instead, and Sudokuno rejects the pattern.

Why it works

The pivot has three candidates, so there are three cases to work through - and the shared candidate is one of them.

If the pivot takes the shared candidate, it is already there. If it takes either of the other two, the wing that shares that digit with the pivot must give it up and is left holding the shared candidate.

In all three cases the shared candidate lands in one of the three cells, so any cell seeing all three cannot hold that digit.

That is the whole difference from a Y-Wing: there, the pivot does not hold the digit and the target only needs to see the two wings. Here the pivot holds it too, so the target must see the pivot as well.

A real example from Sudokuno

The board below is a real Sudokuno puzzle, shown exactly as it stands just before XYZ-Wing applies. Everything up to this point was solved with ordinary techniques.

4 6 7 3 8 1 5 9 2
1
3
6
4 8 7
8 5
7
1 3 6
2 3 8 4 1 7 6 5 9
5 4 6
3 7 1
7
6 5 3 8 2 4
9
6 7 1 3
3
9 7
2 6 5
6 7
1 3
9 4 8

Click away the candidates the technique removes - 1 of them - then press Check. Nothing is saved, and your own game is untouched.

Sudoku: SNC-00605 — Sudokuno

  • Houses: row 7, column 3 and box 7
  • Pivot: R7C3
  • Wings: R7C5 and R9C3
  • Its three candidates: 2, 4 and 5
  • Removes 2 from R7C2
  • Step 33 of 53 in the solve

Key: pivot wing cells target cells

How the hint helps you

Sudokuno's hint reveals the structure in three steps, without giving away the digit straight away.

  1. Hint 1 Find the pivot: a cell with three candidates that sees two cells with two candidates each. Sudokuno highlights the pivot at this stage already.
  2. Hint 2 Each wing uses two of the pivot's three candidates. The wings share one candidate, and their remaining candidates differ.
  3. Hint 3 Whichever candidate the pivot takes, the shared candidate must sit in one of the three cells - so it can be removed from the highlighted cells that see all three, pivot included.

XYZ-Wing is the one technique where the hint highlights a cell on the very first step. The pivot is hard enough to find that it is the right place to begin.

Ready to try it yourself?

You will get a real Sudokuno puzzle at Specialist level S6, where XYZ-Wing is needed to finish it.

Practise XYZ-Wing

What Sudokuno covers

The geometry means an XYZ-Wing never has more than two target cells. Longer variants such as WXYZ-Wing are not implemented.

Where to go next

  • Y-Wing A Y-Wing is three cells that each have exactly two candidates. The pivot sees both wing cells. It shares one candidate with each wing, and the two wings have a third candidate in common. That third candidate has to land in one of the wings, so it can be removed from any cell that sees both of them.
  • Chute Remote Pairs A chute is three boxes in a line - either three side by side or three stacked. If you find two cells in the same chute holding exactly the same candidate pair, and they share no row, column or box, then the three bridge cells in the unused line settle the matter. If one of the pair candidates is missing from the bridge, the other can be removed from the cells that see both ends.

All Sudoku techniques