XYZ-Wing in Sudoku

Specialist S6 Wings

An XYZ-Wing has a pivot with three candidates and two wings with two each. The pivot sees both wings, whose pairs come from its three candidates. The digit shared by the wings must fill one of the three cells, so any other cell seeing all three loses it.

A candidate is a digit that can still go in an empty cell. Cells see each other when they share a row, column or 3×3 box. A strong link means a digit has exactly two possible cells in one such group: exactly one must contain it.

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

For example, let the pivot have 1/2/3 and the wings 1/3 and 2/3. The pivot sees both wings. The shared candidate is 3.

If the pivot is 3, one of the three cells already holds 3. If it is 1, the 1/3 wing must be 3. If it is 2, the 2/3 wing must be 3.

So a cell seeing all three cannot be 3. Unlike Y-Wing, the pivot can hold the shared digit too: seeing only the wings is not enough.

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 a candidate to cross it out; click it again to restore it. Remove the 1 candidates ruled out by this technique, then press Check. This exercise leaves your game unchanged.

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.

The pivot is highlighted at Hint 1. The wings appear at Hint 2, and Hint 3 marks the cells that see all three and can lose the shared candidate.

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 uses three cells with two candidates each. One cell, the pivot, sees both wings. Whichever digit fills the pivot, one wing must hold the digit shared by the wings, so any cell seeing both wings loses that candidate.
  • Chute Remote Pairs Chute Remote Pairs uses two cells with the same candidate pair in one band of three boxes or one stack of three boxes. A third group of cells, the bridge, rules out both endpoints holding one digit. At least one endpoint must hold the other, so cells seeing both lose that other digit.

All Sudoku techniques