Sudoku Wings

A Sudoku Wing is a compact candidate pattern that proves an elimination by showing that a target digit must appear in one of a small set of strategically connected cells.

The best-known Wings — XY-Wing, XYZ-Wing, W-Wing, and WXYZ-Wing — do not all have exactly the same geometry. What connects them is that a small candidate structure behaves like a short logical network: whichever way the unresolved cells resolve, one candidate becomes unavoidable in a position that eliminates the same target elsewhere.

Understanding that shared logic is more useful than memorizing four unrelated pictures.

Quick map of the Wing family

Why Wings are easier to recognize than general Chains

A general Chain can wander through many cells and candidates. A Wing compresses the same kind of implication into a recognizable local shape.

That gives Wings two advantages:

• they are faster for a human to scan for;
• their proof can usually be explained in a few cases rather than a long implication string.

This is why some patterns that can be represented as Chains still deserve their own names.

The common Wing question

For every Wing, ask:

Which candidate is guaranteed to appear in at least one of the relevant endpoint cells?

If a target cell sees every possible place where that candidate is guaranteed to land, the target cannot contain it.

The geometry changes from technique to technique, but the target rule does not.

XY-Wing: the classic three-cell Wing

An XY-Wing uses three bivalue cells:

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

The pivot sees both pincers. If the pivot is X, the YZ pincer becomes Z. If the pivot is Y, the XZ pincer becomes Z.

Therefore one pincer must contain Z, and any cell that sees both pincers cannot be Z.

This is also a short XY-Chain, but the named pattern is easier to recognize.

XYZ-Wing: the target must see the pivot too

XYZ-Wing adds Z to the pivot:

• pivot {X,Y,Z};
• pincers {X,Z} and {Y,Z}.

Now Z may be true in the pivot itself, not only in a pincer. That changes the target condition.

A valid elimination must see:

1. the XZ pincer;
2. the YZ pincer;
3. the XYZ pivot.

Missing the third visibility condition is one of the most common XYZ-Wing mistakes.

W-Wing: the bridge is an external Strong Link

A W-Wing starts with two bivalue cells carrying the same pair {X,Y}.

Those two cells alone prove nothing.

The pattern becomes useful when an external Strong Link on one candidate — say Y — connects the two sides. Whichever end of that Strong Link is true forces one of the bivalue cells to be X.

Therefore at least one endpoint cell is X, and X can be eliminated from any cell that sees both endpoints.

W-Wing is a good reminder that a Wing does not have to use a pivot-and-two-pincer layout.

WXYZ-Wing: four cells, four digits

WXYZ-Wing extends the Wing idea to a four-cell / four-digit structure. The most useful modern interpretation is broader than the old picture of one four-candidate hinge plus three two-candidate wings.

For a Type 1-style WXYZ-Wing, the four cells collectively contain four digits and are restricted to a small set of houses. Exactly one candidate acts as a non-restricted common candidate: not all of its occurrences inside the pattern see one another.

The other digits are restricted within the pattern. This guarantees that the non-restricted digit must be true in at least one of its pattern occurrences.

Any outside candidate that sees all of those occurrences can therefore be eliminated.

The dedicated WXYZ-Wing Guide handles the exact recognition test and why this pattern overlaps conceptually with ALS-XZ.

Wings and Almost Locked Sets

The deeper you go, the less useful it becomes to pretend every Wing is an isolated invention.

XY-Wing, XYZ-Wing and especially WXYZ-Wing can be described through Almost Locked Set reasoning. ALS provides a more general language for many restricted candidate structures that named Wings package into human-recognizable forms.

That does not make the Wing names obsolete. A named pattern can still be much faster to spot than a generic ALS construction.

A practical Wing search order

When candidates are accurate:

1. scan bivalue cells;
2. look for a cell that can act as an XY pivot;
3. test nearby pincers for a common Z;
4. if a trivalue pivot is involved, test XYZ-Wing visibility;
5. look for repeated bivalue pairs that could form W-Wings;
6. only then spend time on larger four-cell Wings.

This order minimizes combinatorial searching.

What makes a Wing invalid?

A pattern fails if its proof does not force the elimination digit somewhere the target can see.

Typical errors include:

• using cells in the wrong houses;
• assuming pincers must see each other in XY-Wing;
• forgetting the pivot visibility condition in XYZ-Wing;
• calling two identical bivalue cells a W-Wing without a Strong-Link bridge;
• counting four cells and four digits as WXYZ-Wing without checking the restricted/non-restricted structure;
• eliminating from a target that does not see every relevant occurrence.

Wings vs subsets

A Naked Triple or Quad is a reservation inside one house.

A Wing often spans more than one house and uses implication or restricted visibility rather than a simple local N-for-N reservation.

Some advanced Wings can also be described using ALS or locked-set language, which is why the boundary becomes less rigid at expert level.

Wings vs Chains

Wings are often short Chains with memorable geometry.

Use the named Wing when the shape is obvious. Use Chain/AIC reasoning when the implication network is easier to follow than the pattern name.

The proof matters more than the label.

FAQ

What is a Wing in Sudoku?

A Wing is a small candidate pattern that proves a common candidate must occur in one of several connected cells, allowing eliminations from cells that see all relevant endpoints.

Which Sudoku Wings should I learn first?

Learn XY-Wing first, then XYZ-Wing and W-Wing. WXYZ-Wing is a later expert extension.

Is XY-Wing the same as Y-Wing?

Yes. Y-Wing is a common alternative name for XY-Wing.

Are Wings really Chains?

Many Wings can be represented as short Chains or ALS relationships. Their separate names are useful because the geometry is easier to recognize directly.

Do all Wing cells have to be bivalue?

No. XY-Wing and W-Wing are bivalue-heavy, but XYZ-Wing has a trivalue pivot and general WXYZ-Wing definitions allow broader candidate distributions.

What to learn next

Study XY-Wing, XYZ-Wing and W-Wing individually, then move to WXYZ-Wing. If the patterns begin to feel like different versions of the same implication idea, continue into XY-Chains, AIC and Almost Locked Sets.