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A W-Wing connects two matching bivalue cells through a Strong Link on one of their candidates.
Suppose the two wing cells are:
- wing A
{X,Y}; - wing B
{X,Y}.
Now find a Strong Link on Y such that:
- one endpoint of the Y link sees wing A;
- the other endpoint sees wing B.
That structure guarantees:
At least one of the two wing cells must be X.
Therefore candidate X can be removed from any cell that sees both wing cells.
The digit used by the Strong Link is the linking digit. The other digit in the matching bivalue cells is the elimination digit.
W-Wing at a glance
Wing A {X,Y}
\
Y Strong Link
/
Wing B {X,Y}The standard pattern needs:
- two bivalue wing cells with the same candidate pair
{X,Y}; - a Strong Link on one of those candidates, here Y;
- one link endpoint connected by visibility to wing A;
- the other link endpoint connected by visibility to wing B;
- at least one candidate X that sees both wing cells.
One endpoint must be Y
The Strong Link can be on X instead. If X is used as the linking digit, Y becomes the elimination digit.
Why W-Wing works
Suppose both wing cells are {5,9} and the external Strong Link is on candidate 9.
Call the two Strong-Link endpoints A and B.
One endpoint is connected to the first wing; the other is connected to the second.
Because A and B form a Strong Link, at least one of them must be 9.
If endpoint A is 9
Wing A sees that 9 and therefore cannot be 9.
Since wing A is {5,9}, wing A must be 5.
If endpoint B is 9
Wing B cannot be 9.
So wing B must be 5.
In every valid outcome, at least one wing cell is 5.
Therefore any candidate 5 that sees both wing cells can be eliminated.
Step-by-step board example
A reliable W-Wing proof sequence is:
- identify two matching bivalue cells;
- choose one of their candidates as the possible linking digit;
- find a Strong Link on that digit;
- confirm one link endpoint sees one wing;
- confirm the other endpoint sees the other wing;
- identify the non-linking wing candidate;
- eliminate that candidate only from common peers of both wings.
The pattern is not proved merely by finding duplicate candidate pairs.
Why matching bivalue cells are not enough
Two remote {5,9} cells do not automatically reserve 5 and 9 between them.
If they do not share a house, they are not a Naked Pair.
Without some additional logical connection, both cells could take the same value or be resolved in unrelated ways.
The Strong Link supplies the missing inference. It ensures that one of the two wing cells must be the non-linking candidate.
That is the essential W-Wing idea:
matching bivalue endpoints + external Strong Link = forced endpoint value.
What counts as the Strong Link?
A Strong Link says that if one endpoint is false for a candidate, the other endpoint must be true.
The easiest W-Wings to recognize use a conjugate pair: a row, column, or box containing exactly two positions for the linking digit.
For example, if a column has only two candidate 9s, those two positions form a Strong Link on 9.
At the pattern-recognition stage, conjugate pairs are usually the cleanest source of W-Wing links.
The later Chains Guides provide the more general language for Strong and Weak Links.
W-Wing as a chain
W-Wing is fundamentally a compact chain pattern.
If the wings are {5,9} and the external Strong Link is on 9, one direction of the inference can be written conceptually as:
wing A is not 5
→ wing A = 9
→ connected 9 endpoint is false
→ other Strong-Link endpoint = 9
→ wing B is not 9
→ wing B = 5So:
if one wing is not 5, the other wing is 5.
The same reasoning works in the opposite direction.
That endpoint relationship proves that at least one wing is 5 and allows common-peer eliminations.
How to find a W-Wing
W-Wing is easiest to search by combining bivalue-cell scanning with Strong-Link scanning.
1. Find repeated bivalue signatures
Look for two cells with the same candidate pair, such as {3,8} and {3,8}.
2. Check whether the cells already share a unit
If they do, test the simpler Naked Pair explanation first.
If they are remote, they become stronger W-Wing candidates.
3. Choose one digit as the possible link digit
For {3,8}, search Strong Links on 3 and then on 8.
4. Test endpoint connectivity
A valid Strong Link must bridge the two wing regions:
- one endpoint sees wing A;
- the other sees wing B.
5. Identify the elimination digit
It is the other candidate in the wing pair.
If the Strong Link is on 8, the elimination digit is 3.
6. Find common peers of the wings
Any external candidate 3 that sees both wing cells can be removed.
7. Verify the two Strong-Link branches
Check that either true link endpoint forces one wing to the elimination digit.
W-Wing vs Naked Pair
The same visual candidate pair can mean two very different things.
Naked Pair
Two {X,Y} cells share one house.
Because those two digits must occupy those two cells, X and Y can be removed from other cells in the shared house.
W-Wing
The two {X,Y} cells are usually remote.
An external Strong Link on X or Y creates the inference between them.
The elimination occurs from cells that see both wings, not simply from every other cell in a shared unit.
If a straightforward Naked Pair already proves the same move, prefer the simpler explanation.
W-Wing vs XY-Wing
Both belong to the Wing family, but the logical architecture differs.
XY-Wing
- three bivalue cells;
- one pivot
{X,Y}; - two pincers
{X,Z}and{Y,Z}; - the pivot directly links the branches.
W-Wing
- two matching bivalue wing cells
{X,Y}; - one external Strong Link on X or Y;
- the external link carries the implication between the wing regions.
There is no XY pivot in a W-Wing.
W-Wing vs Remote Pair
Both can use repeated bivalue candidate pairs and chain-like reasoning.
A standard W-Wing uses two matching bivalue endpoint cells connected through a Strong Link on one digit.
A Remote Pair extends the alternating relationship through a longer chain of cells containing the same two candidates.
That makes W-Wing a useful stepping stone toward Remote Pair and more general chain logic.
Can the two wing cells see each other?
They can, but if two identical bivalue cells share a house, the position should immediately be checked as a Naked Pair.
The name W-Wing is most useful when the external Strong Link is doing necessary logical work that a local subset does not already explain more simply.
This follows a broader editorial principle used throughout these Guides:
Prefer the simplest valid proof of the elimination.
A valid W-Wing can be unproductive
The two wings and Strong Link may form a correct W-Wing structure while no external elimination digit sees both wings.
That means the structure exists but produces no move at the current state.
As with other advanced patterns, always distinguish:
- pattern validity;
- productive elimination.
Common W-Wing mistakes
Treating matching bivalue cells as sufficient
They are only the endpoints. The Strong Link is essential.
Using a Weak Link where a Strong Link is required
The proof needs the guarantee that if one link endpoint is false, the other is true.
Connecting both Strong-Link endpoints to the same wing
The link must bridge the two wing regions.
Eliminating the linking digit
In the standard W-Wing, the elimination is on the other candidate from the wing cells.
Eliminating from a cell that sees only one wing
The target must see both possible forced endpoint locations.
Ignoring a simpler Naked Pair
If the wing cells share a house and the subset proof already works, use it.
A practical W-Wing recognition drill
Whenever you notice repeated bivalue candidate sets, ask two questions:
Do these cells share a unit?
If yes, test Naked Pair.
If no, ask:
Can one of their digits be connected through a Strong Link whose endpoints see opposite wings?
If yes, inspect common peers of the wings for the other candidate.
FAQ
Do W-Wing cells need identical candidates?
In the standard W-Wing taught here, yes. Both wing cells are the same bivalue pair {X,Y}.
Do the two wing cells need to see each other?
No. Remote wing cells are normal. If they do share a unit, check whether a simpler Naked Pair already explains the move.
Which digit is used for the Strong Link?
Either X or Y can be the linking digit, provided a valid Strong Link connects the two wing regions.
Which digit is eliminated?
The wing candidate not used as the Strong-Link digit.
Can a W-Wing eliminate more than one candidate?
Yes. Every candidate of the elimination digit that sees both wing cells can be removed.
Is W-Wing a chain?
Yes. The named pattern is a compact chain structure expressed in an easier-to-recognize form.
What to learn next
Continue into Strong Links, Weak Links, and Sudoku Chains, then revisit W-Wing as one reusable chain template rather than an isolated shape.