The XY-Wing: a pivot and two pincers

What an XY-Wing is

An XY-Wing uses three bivalue cells, meaning cells with exactly two candidates. The middle one, the pivot, has candidates X and Y. The two pincers each see the pivot (they share a row, column or box with it). One pincer has candidates X and Z, the other has Y and Z. Z is the shared digit that does not appear in the pivot.

The result: Z can be removed from every cell that sees both pincers. The pincers do not need to see each other, and they usually do not; in the most useful XY-Wings they sit in different rows, columns and boxes.

Why it works

The pivot must be either X or Y. If the pivot is X, the pincer with X and Z cannot be X (it sees the pivot), so it becomes Z. If the pivot is Y, the pincer with Y and Z cannot be Y, so that pincer becomes Z instead.

There is no third option, so one of the two pincers is always Z. We do not know which one, but a cell that sees both pincers can never be Z: if it were, both pincers would lose their Z and at least one of them would be left with no valid digit. That is why the digit is removed only from cells that see both pincers, not just one.

How to find an XY-Wing

Start from bivalue cells. With pencil marks filled in, look for a cell with two candidates and treat it as a possible pivot, with candidates X and Y. Among the cells it sees, look for another bivalue cell containing X plus one new digit, and a third containing Y plus the same new digit.

The three cells together must use exactly three digits, each cell holding two of them, with the pivot missing Z. Then look at the cells that see both pincers. If any of them still has Z as a candidate, you have a useful XY-Wing.

A practical shortcut: boards late in an expert puzzle often have many bivalue cells. Mark them mentally, then check each one as a pivot. It is slower than scanning for a single digit, but the pattern is small and easy to verify once you see it.

A worked example

On the example board, the pivot is R2C8 (row 2, column 8), with candidates 2 and 6. One pincer is R6C8 in the same column, with candidates 2 and 3. The other pincer is R3C9 in the same box, with candidates 3 and 6. So X is 2, Y is 6 and Z is 3.

If R2C8 is 2, then R6C8 cannot be 2 and must be 3. If R2C8 is 6, then R3C9 cannot be 6 and must be 3. Either R6C8 or R3C9 is 3. Now look at R6C9, with candidates 3 and 5. It shares row 6 with R6C8 and column 9 with R3C9, so it sees both pincers and cannot be 3.

Removing 3 leaves R6C9 with a single candidate: it is 5, shown in gold. In this position nothing simpler worked. The engine found no singles, locked candidates, pairs, triples, X-Wings or Swordfish, so the XY-Wing is the move that keeps the solve going.

Common mistakes

Mismatched pincers. Both pincers must share a different pivot digit: one contains X, the other Y. If both pincers contain X and Z, the pivot being Y forces nothing, and the pattern proves nothing.

Using cells with three candidates. Every cell in an XY-Wing must have exactly two candidates. A pincer with Z plus two other digits can escape to a third digit, breaking the chain. Larger versions exist, such as the XYZ-Wing, but they follow different elimination rules.

Eliminating from cells that see only one pincer. A cell that sees just one pincer might still be Z when the other pincer is the one holding it. Only cells in the overlap of both pincers are safe to clear.

Where the XY-Wing fits

The XY-Wing is a short chain: three cells linked by “if not this, then that” reasoning. Learning it is a good step toward longer chains and patterns like the Skyscraper, which use the same either-or logic with a single digit. It also pairs well with naked triples, since both work with small groups of cells holding few candidates.

On Quietdoku, the XY-Wing is one of the techniques Expert puzzles may require, along with X-Wing and Swordfish. Every puzzle has a unique solution and never requires guessing, and a hint shows the next step and names the technique when you are stuck.