W-Wing

The W-Wing is a pattern that uses two bi-value cells connected by a strong link on one of their shared candidates. It's like a more flexible version of XY-Wing, producing similar eliminations.

What is a W-Wing?

A W-Wing requires:

Two bi-value cells with the same two candidates (e.g., both [3,7])
A strong link on one of those candidates connecting cells that each see one of the bi-value cells
The two bi-value cells don't need to see each other directly

The "W" shape: If you trace the pattern, it somewhat resembles the letter W.

The Structure

Wing Cell A: [3,7]           Wing Cell B: [3,7]
      ↑                            ↑
   sees 7                       sees 7
      ↑                            ↑
   Link Cell X ════════════ Link Cell Y
            (strong link on 7)

Components:

Wing A: Bi-value cell [3,7]
Wing B: Bi-value cell [3,7]
Link X: Cell with candidate 7, sees Wing A
Link Y: Cell with candidate 7, sees Wing B
Strong link: 7 appears only in X and Y within their shared unit

The Logic

Follow the implications:

Case 1: Wing A = 3

A is not 7
This doesn't directly affect the strong link
But at least A = 3

Case 2: Wing A = 7

A = 7
Link Cell X sees 7 (from A)
If X were 7, it would conflict with A. But X might not be 7 anyway...

Wait, let me re-approach:

The strong link says: Either X = 7 or Y = 7 (or both, but in a strong link context, exactly one).

Case: X = 7

Wing A sees X
Wing A ≠ 7 (since X = 7)
Wing A = 3 (bi-value, must be one or the other)

Case: Y = 7

Wing B sees Y
Wing B ≠ 7
Wing B = 3

Conclusion:

Either Wing A = 3, or Wing B = 3
At least one Wing has the digit 3

Elimination: Any cell that sees BOTH Wing A and Wing B can have 3 eliminated!

Why It Works

The strong link ensures that 7 is "covered" by either X or Y. Whichever covers 7, the corresponding Wing must take 3. Since we don't know which, both Wings might have 3, but at least one does. Cells seeing both Wings will always see a 3.

Worked Example

Setup:

        Col 2       Col 5       Col 8
Row 1:                          Wing B [2,6]
Row 3:  X[...6...]               Y[...6...]
                    (strong link: 6 only in X and Y in row 3)
Row 7:  Wing A [2,6]

Components:

Wing A: R7C2 = [2,6]
Wing B: R1C8 = [2,6]
Link cells: R3C2 and R3C8
Strong link: Row 3 has 6 only in columns 2 and 8

Verify visibility:

X (R3C2) sees Wing A (R7C2) via column 2 ✓
Y (R3C8) sees Wing B (R1C8) via column 8 ✓

Logic:

Either X = 6 or Y = 6 (strong link)
If X = 6: Wing A ≠ 6 → Wing A = 2
If Y = 6: Wing B ≠ 6 → Wing B = 2

Elimination:

Either Wing A = 2 or Wing B = 2
Cells seeing both Wings can have 2 eliminated

Find targets:

What sees both R7C2 and R1C8?
R1C2: sees R7C2 (col 2) and R1C8 (row 1) ✓
R7C8: sees R7C2 (row 7) and R1C8 (col 8) ✓

If R1C2 or R7C8 has candidate 2, eliminate it!

Finding W-Wings

Step 1: Find Matching Bi-Value Pairs

Look for two cells with the same two candidates that don't see each other.

Step 2: Look for Strong Links

For one of the shared candidates, find a strong link where:

One end sees the first bi-value cell
Other end sees the second bi-value cell

Step 3: Verify and Eliminate

If the structure is valid, eliminate the other shared candidate from cells seeing both Wings.

W-Wing Variations

Same Box Wings

Wings in the same box, but strong link external:

┌─────────────────┐
│ Wing A [3,8]    │
│                 │
│      Wing B [3,8]│
└─────────────────┘
       ↓   ↓
      Link cells with strong link on 8 outside the box

Cross-Grid Wings

Wings far apart, connected by long-range strong link:

Row 1: Wing A [5,9] ... (empty) ... Link X
Row 5:                              │
Row 9: Wing B [5,9] ... (empty) ... Link Y
       (strong link in column connects X to Y)

Multiple Strong Links Available

Sometimes both candidates have strong links available. Either can form a W-Wing. Choose the one that produces eliminations.

W-Wing vs. XY-Wing

Aspect	XY-Wing	W-Wing
Wing cells	Different candidates (XZ, YZ)	Same candidates
Connection	Direct through pivot	Via strong link
Pivot	Required	Not needed
Flexibility	More rigid structure	More flexible placement

W-Wing advantage: Wings don't need to see each other or a shared pivot.

W-Wing vs. Remote Pairs

Aspect	Remote Pairs	W-Wing
Number of cells	4+ in chain	2 wings + 2 link cells
Structure	Connected chain	Wings connected via strong link
Direct visibility	Each cell sees next	Wings don't need to see each other

Practice Exercise

Find the W-Wing:

R2C4: [4,7]
R8C6: [4,7]
Row 5 has candidate 7 only in C4 and C6

Answer

Pattern:

Wing A: R2C4 = [4,7]
Wing B: R8C6 = [4,7]
Strong link: Row 5 has 7 only in C4 (R5C4) and C6 (R5C6)

Visibility:

R5C4 sees R2C4 via column 4 ✓
R5C6 sees R8C6 via column 6 ✓

Logic:

Either R5C4 = 7 or R5C6 = 7
If R5C4 = 7: Wing A ≠ 7 → Wing A = 4
If R5C6 = 7: Wing B ≠ 7 → Wing B = 4

Elimination: Cells seeing both Wing A (R2C4) and Wing B (R8C6):

R2C6: sees R2C4 (row 2) and R8C6 (col 6)
R8C4: sees R2C4 (col 4) and R8C6 (row 8)

If R2C6 or R8C4 has candidate 4, eliminate it!

Common Mistakes

Mistake 1: Link cells aren't actually a strong link

Verify the connecting cells are the ONLY two with the candidate in their shared unit.

Mistake 2: Wings don't match

Both Wing cells must have the SAME two candidates.

Mistake 3: Link cells don't see Wings

Each link cell must see its corresponding Wing. Same column, row, or box.

Mistake 4: Wrong elimination target

You eliminate the NON-link digit. If the strong link is on 7, eliminate the other digit (3 in our examples).

Quick Reference

W-Wing structure:

Two bi-value cells with same candidates [X,Y]
Strong link on X connecting cells that see each Wing
Wings don't need to see each other

Elimination rule:

Either Wing = Y (the non-link digit)
Eliminate Y from cells seeing both Wings

Finding W-Wings:

Find matching bi-value pairs
Look for strong links on one candidate
Check if link ends see the Wings
Find elimination targets

The formula:

Wings share [A,B]
Strong link on B
Eliminate A from cells seeing both Wings