W-Wing

A W-Wing connects two identical bivalue cells through a Strong Link on one of their candidates.

Suppose the wing cells are:

• wing A {X,Y};
• wing B {X,Y}.

A Strong Link on Y connects the two wing regions so that:

• one endpoint of the Y Strong Link sees wing A;
• the other endpoint sees wing B.

That guarantees:

At least one wing cell is X.

Candidate X can therefore be removed from any cell that sees both wing cells.

Quick pattern

Wing A {X,Y}
       \
        Y Strong Link
       /
Wing B {X,Y}

The Strong Link is on Y.

The elimination is on X, the other wing candidate.

Why W-Wing works

Let the Y Strong Link endpoints be A and B.

Exactly one of those endpoints must contain Y.

If endpoint A = Y

Wing A sees that Y and cannot also be Y.

So wing A = X.

If endpoint B = Y

Wing B cannot be Y.

So wing B = X.

No matter which Strong-Link endpoint is true, one of the two wing cells is X.

Any candidate X that sees both wings is impossible.

Step-by-step board example

Why two identical bivalue cells are not enough

If two {5,9} cells are far apart, nothing automatically connects their values.

They are not a global Pair.

The W-Wing needs the external Strong Link to carry the inference between them.

W-Wing vs Naked Pair

Naked Pair

The two {X,Y} cells share one unit.

That local unit creates the reservation.

W-Wing

The two {X,Y} cells may be remote.

An external Strong Link on one candidate connects the logic.

W-Wing vs XY-Wing

XY-Wing

Three bivalue cells:

• {X,Y};
• {X,Z};
• {Y,Z}.

A pivot connects the pincers.

W-Wing

Two matching bivalue cells {X,Y} plus an external Strong Link on X or Y.

There is no XY pivot.

W-Wing as a chain

If the wing cells are {5,9} and the Strong Link is on 9, the inference is conceptually:

wing A not 5
→ wing A = 9
→ linked 9 endpoint false
→ other 9 endpoint true
→ wing B not 9
→ wing B = 5

So if one wing is not 5, the other wing becomes 5.

That endpoint behavior proves that at least one wing is 5.

How to search for W-Wings

1. Find repeated bivalue signatures

2. Pick one candidate as the possible linking digit

3. Find Strong Links on that digit

4. Test connectivity

One Strong-Link endpoint must see one wing and the other endpoint the other wing.

5. Search common peers of the wing cells

The other candidate is the elimination digit.

What if the wing cells see each other?

If the two identical bivalue cells share a unit, first check the simpler Naked Pair explanation.

A W-Wing label is useful when the remote Strong-Link connection is doing real logical work.

Common mistakes

Using a Weak Link instead of a Strong Link

The proof depends on one link endpoint being forced true if the other is false.

Eliminating the linking digit

The standard elimination is the other wing candidate.

Connecting both Strong-Link endpoints to one wing

The link needs to bridge the two wing regions.

Eliminating from a target that sees only one wing

Invalid.

Treating matching bivalue cells as sufficient

Without the link, no W-Wing.

Recognition drill

When you notice duplicate bivalue candidate sets, do not immediately think Pair.

Ask:

Do these cells share a unit?

If no:

Can one of their digits be connected by a conjugate pair?

FAQ

Do W-Wing cells need identical candidates?

In the standard pattern, yes.

Do they need to see each other?

No.

Which digit is eliminated?

The wing candidate that is not used as the Strong-Link digit.

Is W-Wing a chain?

Yes. The named pattern is a compact chain structure.

What to learn next

Continue into Sudoku Chains, then revisit W-Wing as one short reusable chain template rather than a standalone shape.