XY-Wing

Show the last hint page first

Loading demo...

An XY-Wing is an AIC (alternating inference chain) with a wing structure, formed by three bivalue cells.

How to spot it

Look for three bivalue cells arranged as follows:

• Two cells that cannot see each other contain candidate pairs {x, z} and {y, z}. Their shared candidate is z; these cells are the pincers.
• A third bivalue cell seen by both pincers contains {x, y}, the two candidates that differ between the pincers. This cell is the pivot.
• Once all three cells are in place, candidate z can be eliminated from any cell that sees both pincers.

How it works

• The two candidates in the bivalue pivot form a strong link, so the pivot must contain either x or y.
• Each pincer is also bivalue: the {x, z} pincer must contain either x or z, while the {y, z} pincer must contain either y or z.
• Candidate x in one pincer and candidate y in the other form weak links with their matching candidates in the pivot. If the pivot contains x, the {x, z} pincer must contain z; if the pivot contains y, the {y, z} pincer must contain z.
• Therefore, at least one pincer must contain z, allowing candidate z to be eliminated from any cell that sees both pincers.

Examples

On the board above, R7C7 is the pivot. Candidate 2 in R5C7 forms a weak link with candidate 2 in R7C7, while candidate 5 in R7C3 forms a weak link with candidate 5 in R7C7. Thus, R5C7 and R7C3 are the two pincers. Candidates 2 and 5 play the roles of x and y (the assignment can be reversed without affecting the logic), and candidate 3 plays the role of z. Because R5C3 sees both pincers, candidate 3 can be eliminated from it.

In this example, the pivot and pincers lie in different boxes, and each pincer forms its weak link with the pivot through a shared row or column.

---

On the board above, the pivot shares a box with one pincer, allowing the XY-Wing to eliminate candidate z from multiple cells.