W-Wing in Sudoku

Specialist S8 Wings

A W-Wing connects two cells holding the same candidate pair through a strong link on one digit. This proves that at least one of the pair cells holds the other digit. Remove that other digit from any cell seeing both pair cells.

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

  • Find two cells containing only the same two candidates, for example 7/9. These are the W-Wing cells.
  • Find a strong link on one of those digits, say 7: a row, column or box with exactly two possible cells for 7. These must be two separate cells from the W-Wing cells.
  • One W-Wing cell must see one end of the link, and the other must see the other end. The connections may cross.
  • Find other cells that see both W-Wing cells. They can lose the other digit of the pair, here 9.

Why it works

Use the pair 7/9 with a strong link on 7. Exactly one end of the link must be 7, because those are the only two places for 7 in its row, column or box.

Whichever end holds 7, the W-Wing cell that sees it cannot be 7. That cell has only 7/9, so it must be 9.

At least one W-Wing cell therefore holds 9. We do not know which, but any other cell seeing both cannot be 9.

The digit removed is the other pair digit, not the link digit. Matching pairs alone prove nothing; the strong link and both connections are needed.

A real example from Sudokuno

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

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

Click a candidate to cross it out; click it again to restore it. Remove the 3 candidates ruled out by this technique, then press Check. This exercise leaves your game unchanged.

Example: W-Wing - Sudokuno - not yet published in the puzzle picker

  • Candidate pair 7 and 9
  • Houses: row 9
  • Endpoints: R4C6 and R5C3
  • Strong link: R9C3 and R9C6
  • Removes 9 from R4C1, R5C4 and R5C6
  • Step 40 of 57 in the solve

Key: endpoints strong link 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 The two W-Wing cells are marked. Find their matching candidate pair.
  2. Hint 2 The strong link appears. One W-Wing cell sees one end; the other sees the other end.
  3. Hint 3 The targets are marked. At least one W-Wing cell holds the other pair digit, so cells seeing both lose it.

You only need to know that at least one W-Wing cell holds the other digit. The technique does not tell you which cell it is.

Ready to try it yourself?

You will get a real Sudokuno puzzle at Specialist level S8, where W-Wing is needed to finish it.

Practise W-Wing

What Sudokuno covers

Sudokuno covers the standard W-Wing with two matching candidate pairs and one strong link. Longer chains and grouped variants are outside this form.

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.
  • XYZ-Wing 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.
  • 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.
  • Simple Colouring Simple Colouring follows one digit through a network of strong links. Colour linked cells alternately: either every cell of colour A holds the digit and none of B does, or the reverse. Those two possibilities can rule out candidates.

All Sudoku techniques