W-Wing in Sudoku

Specialist S8 Wings

A W-Wing starts with two cells that have exactly two candidates each, and the same two: say 7 and 9. Somewhere else on the board, one of those digits - say 7 - has exactly two places left in a single row, column or box, and each of the two cells sees one of those two places. That is enough to prove that at least one of the two cells holds 9, so any cell that sees both of them cannot.

How to spot it

  • Look for two cells with exactly two candidates each, holding the same pair. They are usually far apart and cannot see each other.
  • Take one of the two digits and hunt for a strong link on it: a row, column or box with exactly two places left for that digit.
  • Check the connection. One of your two cells must see one end of the link, and the other must see the other end. Either pairing will do - the arms may cross.
  • Now take the other digit of the pair. That is the one you can remove.
  • Find every unsolved cell that still has that digit and sees both of the original cells. All of them lose it, not just the first one you notice.

Why it works

Follow it through with the pair 7 and 9, linked on 7. Suppose the first cell is not 9. Then it must be 7, because it has only those two candidates. The end of the link it sees therefore cannot be 7 - and since the link has only two places for 7, the far end must be 7 instead. The second cell sees that far end, so it cannot be 7 either, which leaves it as 9.

The same argument runs the other way round: assume the second cell is not 9, and you are forced to conclude the first one is. So whichever way you start, at least one of the two cells holds 9.

You never learn which one. You do not need to: a cell that sees both of them sees a 9 whichever cell it turns out to be, so it cannot be 9 itself. That is the whole elimination.

The strong link is what carries the argument across the board. A house with exactly two places left for a digit must hold it in one of them - never both, and never neither - so knowing one end is not the digit tells you the other end is.

Watch the trap: the digit you remove is the *other* one, not the digit the link is built on. Nothing in the argument shows that either cell must be 7, so cells seeing both of them keep their 7 quite legitimately.

Watch the second trap: two cells with the same pair are not a W-Wing on their own. Without a genuine strong link joining them, there is nothing stopping both of them from being 7, and the conclusion collapses.

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 away the candidates the technique removes - 3 of them - then press Check. Nothing is saved, and your own game is untouched.

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 Sudokuno highlights the two cells the technique is built on. Look at their candidates: exactly two each, and the same two. Those are the digits the rest of the argument is about.
  2. Hint 2 Two more cells appear: the strong link. The highlighted unit has only two places left for one of your two digits, so exactly one of them holds it - and each of the first two cells sees one end.
  3. Hint 3 The hint shows which candidates can go. At least one of the two original cells holds the other digit, so every cell seeing both of them loses it.

The hint never says which of the two cells holds the digit, because the technique does not know. Knowing that one of them does is the whole point, and it is already enough to clear the candidates it clears.

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 implements the classical single W-Wing: two cells with the same pair, one strong link, and eliminations on the other digit. The Double W-Wing and the longer Remote Pair chains are not implemented, and neither is the Split Double variant or the grouped form, although other sources discuss them on the same page.

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.
  • XYZ-Wing 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.
  • 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.
  • Simple Colouring Simple Colouring looks at one digit at a time. When a row, column or box has exactly two places left for that digit, the two cells are strongly linked: exactly one of them holds it. Join such pairs together and you get a network that can be coloured alternately in two colours, and then exactly one of the colours is the true one. That is enough to remove candidates, even though you do not know which colour is which.

All Sudoku techniques