# W-Wing

A **W-Wing** is formed by a chain of four cells:

**A**: A cell with two candidates**W**and**X**.**B**: A cell with**X**as a candidate.**C**: A cell with**X**as a candidate.**D**: A cell with two candidates**W**and**X**.

It is required that **A** sees **B**, **C** sees **D**, and **B** and **C** has a strong link on the digit **X**. Then **W** can be eliminated from any cell that is seen by both **A** and **D**.

Suppose **T** is a cell that is seen by both **A** and **D**, and **T** contains **W** as a candidate. Then, the **W-Wing** can be written as follows in Nice Loop notation:

[T]-W-[A]-X-[B]=X=[C]-X-[D]-W-[T] => T<>W

## Example

In this grid, note the **r5c5** and **r6c3** cells with the candidates **2** and **7**. Also, the green **7** candidates form a chain, with **r5c8** and **r6c8** forming a strong link (because **r5c8** and **r6c8** are the only cells with candidate **7** in this box). Hence, we can eliminate the digit **2** from all cells that are seen by both **r5c5** and **r6c3**, i. e., those marked in red.

**Doubly Linked W-Wing Example**

In a doubly-linked W-Wing the identical pairs are linked to 2 digit conjugate pairs rather than a single digit conjugate pair as in the previous example. This is illustrated in the following example where the 17 cells r8c4 and r9c8 are linked to the 7 conjugate pair and the 1 grouped conjugate in box 7. This means that both 1 and 7 must occur either in r8c4 or r9c8. Each of these are conjugate with the 17 conjugates in box 7. Therefore 1 and 7 can be eliminated from other cells in rows 8 or 9. Candidate eliminations are indicated with a -.

Doubly Linked W-Wing Example

|-----------------+-----------------+-----------------| | 5 67 3 | 9 2 68 | 4 178 16 | | 4 167 168 | 5 67 3 | 678 9 2 | | 89 2 689 | 17 4 168 | 678 5 3 | |-----------------+-----------------+-----------------| | 3 4 7 | 2 19 19 | 5 6 8 | | 6 9 2 | 8 5 7 | 1 3 4 | | 1 8 5 | 6 3 4 | 9 2 7 | |-----------------+-----------------+-----------------| | 2 5 4 | 3 1679 169 | 678 178 169 | | 789 16 1689 | 17 -16-79 2 | 3 4 5 | | 79 3 169 | 4 8 5 | 2 17 -169 | |-----------------+-----------------+-----------------|