W-Wing

Two matching pairs linked by a strong link remove a digit.

A W-Wing is a compact chain that links two bivalue cells holding the identical candidate pair {x,y} — cells that do not see one another — through a strong link on one of those two digits found somewhere else on the board. Because that strong link forces one of the matching cells to fall on the other digit, any cell able to see both of them can safely drop that digit. Learning the w-wing sudoku technique gives you a dependable way to keep making eliminations once singles, hidden pairs and box-line reductions have run dry.

Worked example

Two {1,2} cells R1C1 and R5C5 are joined by a strong link on 1 in column 4 (R1C4=R5C4). One of the pair must be 2, so R1C5 — which sees both — loses 2.1217291812
Two {1,2} cells R1C1 and R5C5 are joined by a strong link on 1 in column 4 (R1C4=R5C4). One of the pair must be 2, so R1C5 — which sees both — loses 2.

Here R1C1 and R5C5 are both {1,2} and cannot see each other. Column 4 supplies a strong link on 1: its only 1s sit at R1C4 and R5C4, so exactly one of them must be the 1. If R1C4 is the 1, then R1C1 shares row 1 with it and is forced to 2; if instead R5C4 is the 1, then R5C5 shares row 5 and is forced to 2. Either way, one of R1C1 and R5C5 ends up a 2, so R1C5 — which sees both cells — cannot be 2, and 2 is eliminated from R1C5.

When to use it

Reach for a W-Wing once the grid is peppered with bivalue cells and the easy singles have stalled. Scan for two cells carrying the same pair {x,y} that do not see each other, then hunt for a conjugate pair on x or y whose two ends each line up with one of those matching cells. Once you have that alignment, any cell that sees both matching cells loses the other digit of the pair.

How to use it

  1. Find two matching pairs. Find two cells with the same two candidates, x and y, that do not see each other.
  2. Find the strong link. Find a row, column, or box where x appears in just two cells — one seeing the first pair, the other seeing the second.
  3. Eliminate y. One of the two pairs must be y, so remove y from every cell that sees both of them.
Practice on a real puzzle

Frequently asked questions

What is a W-Wing in Sudoku?

A W-Wing is a short chain built from two bivalue cells that hold the identical candidate pair {x,y} and do not see each other, joined by a strong link on one of those digits. The strong link forces one of the two cells to take the other digit, so any cell that sees both can have that digit removed. In the example above, R1C1 and R5C5 are both {1,2}, and a strong link on 1 in column 4 lets you erase 2 from R1C5.

How do I spot a W-Wing on the grid?

First find two cells with the same pair {x,y} that cannot see each other, then look for a unit — a row, column or box — where one of those digits has only two candidate cells (a strong link), with each end aligned to one of your matching cells. In the example, R1C1 and R5C5 share {1,2}, and column 4's only 1s sit at R1C4 and R5C4, each of which sees one of the pair. Any cell seeing both matching cells, like R1C5, is your elimination target for the other digit.

How is a W-Wing different from an XY-Wing?

An XY-Wing chains three differently-labelled bivalue cells through a shared pivot, using pairs such as {x,y}, {x,z} and {y,z}. A W-Wing instead uses just two cells carrying the identical pair {x,y}, connected not by a pivot but by a separate strong link — a conjugate pair — on one of the digits. So the W-Wing needs two matching cells plus an external strong link, like the 1s in column 4 above, rather than a middle pivot cell.