XYZ-Wing Sudoku Technique

The XYZ-Wing 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.

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

1. Pick a cell with exactly three candidates and treat it as the pivot.
2. Find bivalue cells that see it and whose two digits are both among the pivot's three.
3. Take two such pincers that between them cover the pivot's three digits and share exactly one digit, z.
4. List the cells that see the pivot and both pincers.
5. Delete z from every cell on that list.

A worked example

pivot    r2c2 {1,2,3}
pincer   r2c5 {1,3}     shares row 2 with the pivot
pincer   r3c3 {2,3}     shares box 1 with the pivot

targets  r2c1 {3,5,9}   sees all three cells
         r2c3 {3,6,9}   sees all three cells

The shared digit is 3. Run the three cases: if r2c2 is 3 we are finished; if r2c2 is 1 then r2c5 must be 3; if r2c2 is 2 then r3c3 must be 3. A 3 sits in one of those cells every time.

Now check the targets. Both r2c1 and r2c3 lie in row 2, so they see the pivot and r2c5. Both lie in box 1, so they also see r3c3. They see all three cells, and 3 comes off each of them — leaving {5,9} and {6,9}.

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.