XYZ-Wing

An XYZ-Wing is a three-cell Wing pattern built from:

• pivot {X,Y,Z};
• pincer A {X,Z};
• pincer B {Y,Z}.

Both pincers see the pivot.

The pattern guarantees:

Z is true in the pivot or in one of the two pincers.

Therefore candidate Z can be removed from any cell that sees all three pattern cells.

That final visibility requirement is what most clearly separates XYZ-Wing from XY-Wing.

Quick pattern

Pivot      {X,Y,Z}
Pincer A   {X,Z}
Pincer B   {Y,Z}

Eliminate Z only from common peers of:

• pivot;
• pincer A;
• pincer B.

The three-case proof

Use:

• pivot {2,7,5};
• pincer A {2,5};
• pincer B {7,5}.

Case 1 — Pivot = 2

Pincer A cannot be 2.

So pincer A = 5.

Case 2 — Pivot = 7

Pincer B cannot be 7.

So pincer B = 5.

Case 3 — Pivot = 5

The pivot itself is 5.

In all three cases, at least one of the pattern cells contains 5.

A target that sees all three cannot be 5.

Why the target must also see the pivot

In XY-Wing, the pivot does not contain Z.

So Z must be in one of the two pincers.

In XYZ-Wing, the pivot can itself be Z.

A target that sees only the pincers would fail in the case:

pivot = Z.

Therefore a valid target must see:

• pincer A;
• pincer B;
• pivot.

Step-by-step board example

How to find XYZ-Wings

Start with trivalue cells, not bivalue cells.

1. Choose a pivot {X,Y,Z}

2. Search its peers for a bivalue {X,Z} cell

3. Search for another peer {Y,Z}

4. Identify Z

It is the candidate present in:

• pivot;
• both pincers.

5. Intersect the peer regions

Look only at cells that see all three pattern cells.

6. Verify a target actually contains Z

Without an elimination, the pattern is currently unproductive.

Why XYZ-Wings can be harder to use than XY-Wings

The proof is still short.

Recognition is harder because:

• the pivot has three candidates;
• there are more possible candidate-role assignments;
• target visibility is stricter;
• valid elimination cells may be few.

This makes XYZ-Wing a good example of the difference between logical complexity and recognition complexity.

XYZ-Wing vs Naked Triple

If all three pattern cells share one unit and collectively contain only X/Y/Z, test the simpler Naked Triple explanation first.

Prefer the simplest valid explanation for the current move.

XYZ-Wing vs XY-Wing

Common mistakes

Using a bivalue pivot

That is XY-Wing.

Eliminating from a target that sees only the pincers

That fails when pivot = Z.

Choosing pincers that do not see the pivot

Both relationships are required.

Misidentifying Z

Z is the candidate shared across all three pattern cells.

Assuming every valid XYZ structure produces an elimination

Check common peers.

Recognition drill

Find a trivalue cell.

Then ask:

Can two of its peers each drop one different pivot candidate while sharing the same third candidate?

If yes, check the three-way common-peer region.

FAQ

Which candidate is eliminated?

Z.

Why must the target see the pivot?

Because the pivot itself may be Z.

Are both pincers bivalue?

In the standard XYZ-Wing taught here, yes.

Is XYZ-Wing always harder than XY-Wing?

Not necessarily, but the stricter geometry often makes it harder to recognize.

What to learn next

Move into Strong and Weak Links, then return to W-Wing, whose proof depends directly on a Strong Link.