XY-Wing (Y-Wing) in Sudoku: Pivot and Pincers Explained

An XY-Wing (also called a Y-Wing) uses three cells with exactly two candidates each. One cell, the pivot, holds X and Y. It sees two other cells, the pincers: one holds X and Z, the other holds Y and Z. Whatever the pivot turns out to be, one of the pincers must be Z. So any cell that sees both pincers cannot be Z. It is usually the first advanced technique people learn after the X-Wing, and it shows up more often than the X-Wing on hard puzzles.

The pattern

• Pivot: a cell with exactly two candidates, X and Y.
• Pincer 1: a cell with exactly X and Z that shares a row, column, or box with the pivot.
• Pincer 2: a cell with exactly Y and Z that also shares a row, column, or box with the pivot.

The pincers do not need to see each other. They usually do not, which is why the pattern is useful.

Why it works

The pivot is X or Y. There is no third option.

• If the pivot is X, pincer 1 (X or Z) cannot be X, so it is Z.
• If the pivot is Y, pincer 2 (Y or Z) cannot be Y, so it is Z.

One of the two pincers is Z in every possible solution. A cell that sees both pincers would clash with whichever one is Z, so Z can be erased from it. You do not learn which pincer is Z, and you do not need to.

Worked example

Say R1C1 has candidates {3, 7}. That is the pivot, with X = 3 and Y = 7.

• R1C5 has {3, 9}. It shares row 1 with the pivot. Pincer 1, with Z = 9.
• R2C3 has {7, 9}. It shares box 1 with the pivot. Pincer 2, also with Z = 9.

If R1C1 is 3, R1C5 is 9. If R1C1 is 7, R2C3 is 9. Either way, a 9 lands on R1C5 or R2C3.

Which cells see both? R2C4, R2C5, and R2C6 share box 2 with R1C5 and row 2 with R2C3. R1C2 and R1C3 share row 1 with R1C5 and box 1 with R2C3. Remove 9 from any of those five cells that still has it as a candidate.

How to find XY-Wings

1. Fill in full pencil marks. You need accurate candidates in every cell. The pattern is invisible without them. See our pencil marks guide.
2. List the bivalue cells. Only cells with exactly two candidates can take part. On a hard puzzle there are often ten to twenty of them.
3. Try each one as the pivot. For pivot {X, Y}, look among the cells it sees for a bivalue cell containing X (but not Y) and another containing Y (but not X). Check that their other digit is the same.
4. Find the overlap. Mark the cells that see both pincers and check them for Z.

Where to look for eliminations depends on the shape. If one pincer is in the pivot's row and the other in its column, and all three are in different boxes, only one cell sees both pincers: the fourth corner of the rectangle. If one pincer shares the pivot's box, like the example above, the overlap is up to three cells in each of two short segments.

XY-Wing vs XYZ-Wing

An XYZ-Wing is a close relative where the pivot has three candidates, X, Y, and Z. Because the pivot itself might be Z, the only cells you can clear are those that see all three cells. That usually means cells in the pivot's box. Learn the XY-Wing first. The XYZ-Wing makes sense once the XY-Wing is second nature.

The Dojo's advanced drills, including this one, are taken from real graded puzzles at the moment the technique is the next step, with pencil marks filled in. Other patterns at this level: X-Wing, Swordfish, and simple coloring.