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 solving 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.
