XYZ-Wing

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

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

Both pincers see the pivot.

The important conclusion is:

Z must be 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 three-way visibility requirement is the defining practical difference between XYZ-Wing and XY-Wing.

XYZ-Wing at a glance

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

For the standard pattern:

• the pivot has exactly three candidates X, Y, and Z;
• both pincers are bivalue;
• both pincers see the pivot;
• Z appears in the pivot and both pincers;
• a valid elimination target contains Z;
• the target sees the pivot and both pincers.

The letters describe candidate roles. They can represent any three distinct Sudoku digits.

Why XYZ-Wing works

Use this concrete pattern:

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

The pivot must eventually be 2, 7, or 5.

If the pivot is 2

Pincer A sees the pivot and cannot also be 2.

So pincer A must be 5.

If the pivot is 7

Pincer B sees the pivot and cannot also be 7.

So pincer B must be 5.

If the pivot is 5

Then Z is true in the pivot itself.

Those are all possible pivot values.

Therefore, in every solution, at least one of these three cells is 5.

Any candidate 5 that sees all three cells is impossible.

Why the target must see the pivot

This is the check that prevents most invalid XYZ-Wing eliminations.

In an XY-Wing, the pivot is {X,Y} and does not contain Z. So Z must be true in one of the pincers.

In an XYZ-Wing, the pivot is {X,Y,Z}. One valid branch is:

pivot = Z.

If a proposed target sees the pincers but not the pivot, that target would not conflict with this branch.

So the target must see:

• pincer A;
• pincer B;
• the pivot.

A useful recognition habit is to identify the three-way peer intersection before making any elimination.

Step-by-step board example

A reliable proof sequence is:

1. identify the trivalue pivot;
2. label its candidates X, Y, and Z;
3. find an {X,Z} pincer that sees the pivot;
4. find a {Y,Z} pincer that sees the pivot;
5. confirm Z is common to all three pattern cells;
6. intersect the peer regions of pivot + both pincers;
7. eliminate Z only from targets inside that intersection.

If a tempting target sees only the two pincers, reject it.

Do the pincers need to see each other?

No general pincer-to-pincer visibility requirement is needed for the standard proof.

Each pincer must see the pivot because that is what makes the X and Y branches force Z.

The elimination target, however, must see all three cells because Z may be true in any one of them.

As with XY-Wing, the technique is defined by logical relationships rather than by a fixed drawing.

How to find an XYZ-Wing

XYZ-Wing is naturally searched cell-first, but unlike XY-Wing the best starting cells are trivalue cells.

1. Find a promising {X,Y,Z} pivot

A trivalue cell is only a candidate pivot. Most trivalue cells do not form XYZ-Wings.

2. Search its peers for bivalue subsets

Look for a cell {X,Z}.

Then look for another pivot peer {Y,Z}.

3. Identify the common candidate Z

Z is the digit that appears in:

• the pivot;
• pincer A;
• pincer B.

4. Find the three-way visibility region

A target must see all three pattern cells.

This is usually much smaller than the two-pincer intersection used by XY-Wing.

5. Check for target candidates Z

If the common-peer region contains no Z, the pattern has no productive elimination.

6. Verify all three pivot cases

Test pivot = X, pivot = Y, and pivot = Z.

The elimination is valid only if every case makes Z true in a cell seen by the target.

Why XYZ-Wing is often harder to recognize

The logical proof is short, but recognition can be significantly harder than XY-Wing.

Several factors contribute:

• the pivot has three candidates rather than two;
• there are more possible assignments of X, Y, and Z;
• the useful peer intersection is smaller;
• a target must see three pattern cells instead of two;
• many visually plausible structures have no actual elimination.

This is a good example of the difference between proof complexity and recognition complexity.

A technique can have an easy proof but still be difficult to spot on a crowded candidate grid.

XYZ-Wing geometry

There is no single required orientation.

The pivot may see one pincer by row and another by box, by column and box, or through another valid combination of Sudoku houses.

Instead of memorizing a shape, verify:

1. candidate roles;
2. pivot-to-pincer visibility;
3. three-way target visibility.

Those three checks define the pattern more reliably than its appearance.

XYZ-Wing vs XY-Wing

The extra Z in the XYZ pivot is not a cosmetic difference. It is exactly why the elimination region becomes stricter.

XYZ-Wing vs Naked Triple

Both may use three cells whose combined candidates are X, Y, and Z.

Naked Triple

All three cells share one house, and those three digits are reserved for those three cells.

The eliminations occur elsewhere in that shared house.

XYZ-Wing

The proof relies on the pivot-and-pincers implication structure and on the fact that Z must be true in one of the three pattern cells.

If all three cells lie in one unit and form a valid Naked Triple, prefer the simpler subset explanation.

XYZ-Wing vs WXYZ-Wing

A WXYZ-Wing is a broader four-digit Wing-family structure that may be encountered in more expert solving systems.

It is not simply an XYZ-Wing with one extra random candidate.

For the current corpus, the useful boundary is:

• XYZ-Wing = standard three-cell {XYZ} + {XZ} + {YZ} pattern;
• more complex expanded Wings belong to later expert expansion.

This keeps the current Guide precise without pretending the Wing family ends here.

Common XYZ-Wing mistakes

Starting with a bivalue pivot

That is XY-Wing territory.

Using pincers that do not see the pivot

Both pincer-to-pivot peer relationships are required.

Eliminating from a target that sees only the pincers

That fails when the pivot itself becomes Z.

Misidentifying Z

Z is the candidate shared by all three pattern cells.

Assuming the pincers must see each other

The standard proof does not require that relationship.

Treating every {XYZ} pivot with two nearby bivalue cells as a Wing

The candidate-role pattern must be exactly compatible with {XZ} and {YZ}.

Ignoring the possibility that there is no target

A valid structural pattern can be unproductive if no external Z sees all three cells.

A practical XYZ-Wing recognition drill

Choose a trivalue cell {X,Y,Z}.

Then ask:

Can I find one peer that removes Y from the pivot set, leaving {X,Z}, and another that removes X, leaving {Y,Z}?

If yes, ask:

Is there any candidate Z that sees the pivot and both pincers?

That final question separates a useful XYZ-Wing from a merely interesting arrangement.

FAQ

Which candidate is eliminated in an XYZ-Wing?

Z, the candidate shared by the pivot and both pincers.

Why must the target see the pivot?

Because the pivot itself may be Z. A target that cannot see the pivot would not be contradicted in that branch.

Are both pincers bivalue?

In the standard XYZ-Wing taught here, yes: {X,Z} and {Y,Z}.

Do the pincers have to see each other?

No. Both must see the pivot; the elimination target must see all three pattern cells.

Can XYZ-Wing produce several eliminations?

Yes. Every candidate Z in the common peer region of all three pattern cells can be removed.

Is XYZ-Wing always harder than XY-Wing?

Not necessarily. Its proof is still short, but its stricter geometry and trivalue pivot often make it harder to recognize.

What to learn next

Move into Strong Links and Weak Links, then study W-Wing, where a Strong Link connects two matching bivalue cells instead of using an XY pivot.