XYZ-Wing

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An XYZ-Wing is an extension of the XY-Wing. Its defining feature is that the pivot cell of the XY-Wing also contains candidate $Z$. This breaks the strict alternating-chain rule, but eliminations still work. The elimination candidate must now form weak links not only with candidate $Z$ in both pincers, but also with candidate $Z$ in the pivot cell.

The logic is simple. The proposition “$Z$ is in the pivot cell” can be either true or false. If it is true, then any candidate $Z$ that forms a weak link with candidate $Z$ in the pivot can be eliminated immediately. If it is false, the structure reduces to a standard XY-Wing, and the same candidate $Z$ can still be eliminated. So regardless of whether the pivot finally takes $Z$, the target $Z$ is excluded.

For example, in the board below, take $R4C2$ as the pivot cell and $R4C3$ and $R9C2$ as the pincers. We can first identify an XY-Wing where $X$ and $Y$ are $2$ and $7$, and $Z$ is $3$. But the pivot cell $R4C2$ also contains candidate $3$, so the structure becomes an XYZ-Wing. Candidate $3$ in $R5C2$ forms weak links with candidate $3$ in each of $R4C2$, $R4C3$, and $R9C2$, so it can be eliminated.

Because an XYZ-Wing requires the elimination candidate to form weak links with candidate $Z$ in the pivot as well as in both pincers, it can occur only when the pivot shares a box with one of the pincers. That same condition also means the pattern can sometimes eliminate more than one candidate, as in the next example.