XYZ-Wing
XYZ-Wing is like XY-Wing, but the pivot has all three candidates: XYZ.
Learn the techniqueXY-Wing
XY-Wing uses three cells with two candidates each: a pivot with candidates XY and two wings with XZ and YZ.
See the technique in practice
Work through the examples step by step. Each step explains what you see on the puzzle and why the conclusion holds.
Example:
- We look at the cell at row 9, column 3. It has only two candidates left, 5 and 9. We call this cell the pivot.
- Look at the neighboring cell at row 9, column 6. It shares digit 5 with the pivot and has 3 as the only other candidate.
- Look at the other neighboring cell at row 9, column 7. It shares digit 9 with the pivot and also has 3 as the only other candidate.
- The pivot must become either 5 or 9. If it becomes 5, the neighbor cell with the same digit cannot also have 5, so it is forced to become 3. If the pivot becomes 9 instead, the other neighbor is forced the same way to become 3. At least one of the two neighbors therefore always contains 3.
- Any cell that sees both neighbors therefore cannot have 3 either. We remove it from the notes in row 9, column 5 and row 9, column 9.
How to recognize the pattern
XY-Wing is built on three cells that each have exactly two candidates. The midpoint is called the pivot and has candidates X and Y. The two wings see the pivot and share each their candidate with it: one wing has X and Z, the other has Y and Z. Digit Z is shared between the wings, and it is Z that can be removed.
The logic is a short if-then chain. If the pivot becomes X, the wing with X and Z is forced to become Z. If the pivot becomes Y, the other wing is forced the same way to become Z. One of the wings therefore becomes Z regardless, so all cells that see both wings can lose Z. Look for cells with two candidates in the same area, because three such cells close to each other are often candidates for the pattern.
Step-by-step procedure
- Mark all cells with exactly two candidates.
- Choose one of them as the pivot and call the candidates X and Y.
- Look for two cells that see the pivot, where one has X and Z and the other has Y and Z.
- Find the cells that see both wings, and remove Z from them.
Common mistakes
- Removing Z from cells that only see one wing. The removal requires the cell to see both wings at the same time.
- Using a pivot with three candidates. Then you are into XYZ-Wing, which has stricter requirements for where the removals apply.
- Removing Z from the pivot without further thought. The pivot has no Z among its candidates in a true XY-Wing, and if it does, the pattern is misidentified.
When do you need the technique?
Puzzles at Extreme level require techniques that combine three or more cells in logical reasoning of the if-then type. Try your way through the examples below, step by step, using the same tools the solver uses on your own puzzle.
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