W-Wing Sudoku Technique: Two Matching Pairs and a Strong Link

A W-Wing is two cells with the same two candidates, say X and Y, that do not see each other, joined by a strong link on X: a row, column or box where X has only two spots, one seeing each cell. One of the two cells must then be Y, so any cell that sees both can lose Y.

To spot one, find two matching bivalue cells that are far apart, then look for a line where one of their digits appears exactly twice, with one end next to each cell. It belongs to the same family as the XY-Wing.

How the W-Wing works

In this position from a real library puzzle, R1C7 and R8C9 both have exactly 4 and 5 as candidates. They share no row, column or box, so a naked pair does not apply.

Column 3 has only two cells that can hold a 4: R1C3 and R8C3. One of them is the 4 of column 3. Follow each case:

• If R1C3 is 4, then R1C7, in the same row, cannot be 4. It must be 5.
• If R8C3 is 4, then R8C9, in the same row, cannot be 4. It must be 5.

So at least one of R1C7 and R8C9 is a 5. A cell that sees both of them cannot be 5. There are four such cells with a 5 candidate:

• R1C9 is in row 1 with R1C7 and column 9 with R8C9.
• R2C9 and R3C9 are in box 3 with R1C7 and column 9 with R8C9.
• R8C7 is in column 7 with R1C7 and row 8 with R8C9.

All four lose their 5. That is a lot for one pattern. The two wing cells are in different bands and stacks, so the houses around them cross in several places.

Puzzle 201190 from the Sudoku Royale library. Givens (row by row, dots are blanks): .....8.2..1..9.6..6.....3....5.197.8.4.8.3.56.6...21...735.4...12................. The candidates shown are what is left after singles, pointing and claiming, naked and hidden pairs, triples and quads, X-Wing and Swordfish, with nothing else applied.

How to spot a W-Wing

1. List the bivalue cells. Only cells with exactly two candidates can be the ends.
2. Find matching pairs that do not see each other. Two cells with the same two digits in the same house would be a naked pair. The W-Wing needs them apart.
3. Look for a strong link on one of their digits. Check the rows, columns and boxes for a house where that digit has only two cells, one seeing each wing cell. The link cells must not be the wing cells themselves.
4. Eliminate the other digit. If the link is on X, remove Y from every cell that sees both wing cells.

Try both digits of the pair. In the example the link is on 4 and the eliminations are 5s. A strong link on 5 would instead remove 4s.

W-Wing, XY-Wing and XY-Chain

The XY-Wing uses three bivalue cells. The W-Wing uses two, with a strong link doing the job of the pivot. Written as a chain it reads: 5 or 4 in R1C7, 4 in R1C3, 4 in R8C3, 4 or 5 in R8C9. That is a short XY-Chain with one strong link on a single digit in the middle. The middle step, a strong link on one digit, is the building block of an X-Chain.

Related wing patterns are the XYZ-Wing and, for a general version with groups of cells, almost locked sets. For the order to learn them in, see our list of sudoku techniques in learning order or the overview of advanced sudoku strategies.