XYZ-Wing in Sudoku: How It Works and How It Differs From XY-Wing

An XYZ-Wing is three cells: a pivot with exactly three candidates X, Y and Z, and two pincers the pivot can see, one holding X and Z, the other Y and Z. One of the three cells must be Z, so any cell that sees all three can lose Z.

To spot one, look for a cell with three candidates, then check the cells it sees for two-candidate cells whose digits both come from the pivot and share one digit. The eliminations are rare because a cell has to see all three, which in practice means it sits in the pivot's box and in a line through the pivot.

How the XYZ-Wing works

In the example, from a real puzzle in our library, the pivot is R2C2 with candidates 7, 8 and 9. It sees two cells with two candidates each:

• R3C1 holds 8 and 9. It shares box 1 with the pivot.
• R5C2 holds 7 and 9. It shares column 2 with the pivot.

The shared digit, Z, is 9. Now check what the pivot can be:

• If R2C2 is 7, then R5C2 cannot be 7, so R5C2 is 9.
• If R2C2 is 8, then R3C1 cannot be 8, so R3C1 is 9.
• If R2C2 is 9, the pivot itself is the 9.

In every case one of the three cells is a 9. A cell that sees all three would clash with that 9. The only such cell with a 9 candidate is R3C2: it is in box 1 with the pivot and R3C1, and in column 2 with the pivot and R5C2. R3C2 goes from 2, 7, 8, 9 down to 2, 7, 8.

Puzzle 201096 from the Sudoku Royale library. Givens (row by row, dots are blanks): .5....2....6.2..13....1.46..4.6...3.1.5..268...35...4.2.4396...7.................. 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 an XYZ-Wing

1. Start with three-candidate cells. Each one is a possible pivot. On a stuck hard puzzle there are usually only a handful.
2. Look for bivalue cells the pivot sees. You want one cell holding two of the pivot's digits and another holding a different two, with one digit in common.
3. Check that Z is shared. The digit both pincers have is Z. The pivot must have it too, which it always will if both pincers use only pivot digits.
4. Find cells that see all three. One pincer almost always sits in the pivot's box and the other in the pivot's row or column. The targets are the cells of that box on that line, at most two of them besides the pivot.

XYZ-Wing vs XY-Wing

The XY-Wing uses a pivot with two candidates, X and Y. The pivot cannot be Z, so the 9 in the example would have to be in one of the two pincers, and any cell seeing both pincers loses Z. That usually gives more targets.

The XYZ-Wing adds Z to the pivot. Now the pivot might be the Z itself, so a target has to see the pivot as well. That extra condition is what shrinks the eliminations to one or two cells. If you already look for XY-Wings, you are checking the right cells: when a would-be pivot has a third candidate that matches the pincers' shared digit, you have an XYZ-Wing.

Both are small cases of the ALS-XZ rule. In the XYZ-Wing, the pivot and one pincer together form an almost locked set: two cells with three candidates. Longer versions of the same idea are XY-Chains and the W-Wing.

For the order to learn these in, see our list of sudoku techniques in learning order or the overview of advanced sudoku strategies.