The pivot with three candidates

The short version

An XYZ-Wing is an XY-Wing with a fatter pivot. The pivot holds {X, Y, Z} instead of {X, Y}, and both pincers hold {X, Z} and {Y, Z}. All three cells sit in the same box.

Whichever of the three values the pivot takes, one of the three cells ends up as Z. So any cell that sees all three cells cannot be Z.

The shape

• Pivot: {X, Y, Z} — three candidates.
• Pincer A: {X, Z}.
• Pincer B: {Y, Z}.
• All three cells in one common box.
• Eliminate Z from cells that see the pivot and both pincers.

That last line is where this technique differs from its sibling, and it is the part solvers get wrong. In an XY-Wing you only need to see the two pincers. Here you must see all three cells, because the pivot itself can be Z.

Why it works

Three cases, one per pivot candidate.

• Pivot is X → pincer A {X, Z} cannot be X (same box) → pincer A is Z.
• Pivot is Y → pincer B {Y, Z} cannot be Y → pincer B is Z.
• Pivot is Z → the pivot itself is Z.

Every branch puts a Z somewhere in the pattern. The three cells cannot all avoid Z.

So any cell that sees all three has no room to be Z: if it were, all three pattern cells would be barred from Z, and none of the three cases would be possible.

Note what the third case costs you. In an XY-Wing, the pivot was never able to be Z — Z was the one digit absent from the pivot. Here the pivot can take Z, so a cell that sees only the two pincers is not restricted: if the pivot is Z, both pincers are free.

A real board

All three cells sit in the bottom-left box:

• R8C3 holds {4, 5, 8} — the pivot.
• R7C3 holds {4, 5} — pincer A.
• R9C3 holds {4, 8} — pincer B.

Z is 4, present in both pincers and in the pivot.

• If the pivot is 5, then R7C3 can no longer be 5, so R7C3 is 4.
• If the pivot is 8, then R9C3 can no longer be 8, so R9C3 is 4.
• If the pivot is 4, then the pivot is 4.

Every branch places a 4 inside the pattern. Now consider R7C1 and R9C1 — both hold 4, and both sit in the same box as all three pattern cells. They see the pivot, R7C3 and R9C3. So neither can be 4, and both lose it.

Two eliminations, from three cells, and the puzzle moves.

How to scan for one

1. Start from three-candidate cells. Every pivot here has exactly three, so the candidate pool is small and easy to list.
2. Look for its two pincers within the same box. Pincer A must be {X, Z} for two of the pivot's three values, and pincer B {Y, Z} for the third pairing. The pivot's three values split as one shared pair plus two singles.
3. Confirm the box. All three cells must share a unit, and in practice it is always a box — rows and columns are too wide for the geometry to work out.
4. Find cells that see all three. This is the expensive step, and the reason to only check XYZ-Wings when the board is genuinely stuck. For each candidate cell containing Z, verify it shares a unit with the pivot and with both pincers.
5. Delete sparingly. Fewer cells qualify than in an XY-Wing. If you find yourself eliminating from half the box, you have made an error.

XYZ-Wing vs XY-Wing

The XYZ-Wing is strictly weaker: fewer patterns qualify, and each pattern eliminates from fewer cells. It earns its place because puzzles exist where the three-candidate pivot is the only structure available.

There is a naming trap worth knowing. "XYZ" here describes the pivot's candidates, not a three-cell rectangle — the pattern is still three cells, not four. Confusing it with an X-Wing or a unique rectangle is common among solvers who learn the names before the shapes.

XY-Wing, the two-candidate version.

The traps

Eliminating from cells that see only the pincers. The single most common error, and it produces invalid deletions that break the board later. The pivot can be Z, so the pivot must be in view.

Cells in a different box. If the three pattern cells are not all in one box, the pattern is not an XYZ-Wing at all. The forcing depends on mutual visibility between the pivot and both pincers, which the box supplies.

Mismatched Z. Pivot {4, 5, 8} with pincers {4, 5} and {5, 8} has no common Z that appears in both pincers and leaves the pivot free to take a third value. Check that Z is in both pincers, and that the pivot holds Z too.

Treating the elimination as a placement. Like every wing, this removes a candidate. The follow-up — usually a naked single in the affected cell — is the actual payoff.

Where to use it

XYZ-Wings appear in hard and expert puzzles, often in the same boards where XY-Wings fail to materialise. They are worth knowing for completeness rather than as a everyday tool: the search cost is high and the yield is one or two candidates.

The realistic approach is to check them last among the wings, after XY-Wings and skyscrapers, because those are cheaper to find and usually fire first on the same boards.