XYZ-Wing Sudoku Technique
The XYZ-Wing sudoku technique is a Y-Wing with a fatter hinge. Instead of a two-candidate pivot it uses a three-candidate one, and the price of that extra digit is a much smaller elimination zone, but the pattern turns up often enough to be worth learning.
What the pattern is
Three cells are involved:
- A pivot with exactly three candidates,
{x,y,z}. - A pincer seeing the pivot, with exactly two candidates
{x,z}. - A second pincer seeing the pivot, with exactly two candidates
{y,z}.
The digit z appears in all three cells. Whichever value the pivot takes, one of the three cells ends up holding z:
- Pivot is z, and we are done.
- Pivot is x, so the first pincer must be z.
- Pivot is y, so the second pincer must be z.
So z is guaranteed somewhere in those three cells. Any cell that sees all three loses z.
| Y-Wing | XYZ-Wing | |
|---|---|---|
| Pivot | Two candidates | Three candidates |
| Pincers | Two, both bivalue | Two, both bivalue |
| Cells that must see the pattern | The two pincers | The pivot and both pincers |
| Usual targets | Several cells | One or two cells, normally inside the pivot's box |
Because the pivot itself can hold z, the target has to see the pivot too. That is why an XYZ-Wing almost always fires inside the box that contains the pivot.
When to look for it
- You have several three-candidate cells left and the bivalue cells around them share digits.
- A Y-Wing search failed only because the hinge had one candidate too many.
- Two bivalue cells in a box or line overlap on exactly one digit, and a nearby three-candidate cell contains both of their other digits.
How to apply it, step by step
- Pick a cell with exactly three candidates and treat it as the pivot.
- Find bivalue cells that see it and whose two digits are both among the pivot's three.
- Take two such pincers that between them cover the pivot's three digits and share exactly one digit, z.
- List the cells that see the pivot and both pincers.
- Delete z from every cell on that list.
A worked example
Watch the three cells: the pivot A8 holds 5, 6 and 7, the pincer A5 (row A) holds 6 and 7, and the pincer C7 (box 3) holds 5 and 6. The shared digit is 6.
Run the three cases. If A8 is 7, A5 loses its 7 and must be 6. If A8 is 5, C7 loses its 5 and must be 6. If A8 is 6, the pivot holds it. A 6 sits in one of A8, A5 or C7 every time.
Now check the targets. A7 (5, 6, 9) shares row A with A8 and A5, and box 3 with A8 and C7, so it sees all three cells. It is the only such cell, and 6 comes off it, leaving 5 and 9.
Common mistakes
- Forgetting the pivot when checking visibility. A target that sees both pincers but not the pivot proves nothing, because the pivot could be z itself.
- Using a pincer with three candidates. Both pincers must be strictly bivalue.
- Picking pincers that share two digits. They must overlap on exactly one, and together with the pivot cover exactly three digits.
- Requiring the pincers to see each other. They do not have to.
- Eliminating a digit other than z. Only the shared digit is at risk.
Where you meet it
XYZ-Wings show up on hard classic Sudoku, usually right after the simpler wings stop working. They are a good habit to build before you start on chains, because the same pivot-and-pincer reading is what makes longer chain patterns readable later on.