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# WXYZ-Wing

The WXYZ-Wing is an extension of the XYZ-Wing, and is sometimes called XYZW-Wing. Seeing the extension of XYZ-Wing to WXYZ-Wing, one can further extend the technique to VWXYZ-Wing, UVWXYZ-Wing and so on.

## Simplest form of the WXYZ-Wing

```.-----------.----------.----------.
| *  *  WXYZ| .  XZ .  | YZ .  .  |
| .  WZ .   | .  .  .  | .  .  .  |
| .  .  .   | .  .  .  | .  .  .  |
:-----------+----------+----------:
```

Similar to the XYZ-Wing, the pivot has the candidates WXYZ. In all possible options for the pivot cell, Z will be eliminated from the starred cells.

## Extended form of the WXYZ-Wing

The idea from the extended form of the WXYZ-Wing comes from the following observation:

• If the WZ is W, then the WXYZ (which becomes XYZ), XZ and YZ cells form a naked subset, and so Z can be eliminated from the starred cells.
• If the WZ is Z, then obviously Z can also be eliminated from the starred cells.

Put another way, observe that there are two almost locked sets: (a) WXYZ, XZ and YZ; and (b) WZ. Then the ALS-XZ rule can be applied with W being the restricted common, so Z can be eliminated from the starred cells.

### Type 1

```.-----------.----------.----------.
| *  *  WXYZ| .  XYZ.  | XYZ.  .  |
| .  WZ .   | .  .  .  | .  .  .  |
| .  .  .   | .  .  .  | .  .  .  |
:-----------+----------+----------:
```

We can perform the same eliminations for Z on the starred cells so long:

• The WXYZ and the two XYZ cells form an almost locked set.
• The WXYZ and WZ cells belong to the same box.
• Each of the four cells may contain a subset of the candidates as shown, e.g. the WXYZ cell may contain only XZ.
• One of the XYZ cells may appear in the same box as WZ, in which case the XYZ cell may also have W as a candidate.

### Type 2

```.-----------.----------.----------.
| *  *  WXYZ| .  .  .  | WZ .  .  |
| XYZ.  .   | .  .  .  | .  .  .  |
| .  XYZ.   | .  .  .  | .  .  .  |
:-----------+----------+----------:
```

We can perform the same eliminations for Z on the starred cells so long as:

• The WXYZ and the two XYZ cells form an almost locked set.
• The WXYZ and WZ cells belong to the same line.
• Each of the four cells may contain a subset of the candidates as shown, e.g. the WXYZ cell may contain only XZ.
• One of the XYZ cells may appear in the same line as WZ, in which case the XYZ cell may also have W as a candidate.

## Example

This is a Type 1 WXYZ-Wing (marked yellow) which leads to the elimination of 4 from the blue cell.

## WXYZ-Wing as an ALS-XZ

The WXYZ-Wing can be replicated as an ALS-XZ move by considering the XZ cell as an ALS and the other three cells as the other ALS, with W as a restricted common.