Wings
XYZ-Wing
An XYZ-Wing is an XY-Wing with a pivot that carries three candidates instead of two. The pivot reads {X, Y, Z}, and the two wings are {X, Z} and {Y, Z}. Whatever the pivot turns out to be, one of the two wings is forced to Z, so Z can be removed from any cell that sees the pivot and both wings at once.
- Level
- Advanced
The pattern
What it does.
Take an XY-Wing and give the pivot a third candidate. The argument survives, but the elimination shrinks: because the pivot itself might be Z, Z can only be removed from cells that see the pivot as well as both wings.
That's a smaller set of cells than an XY-Wing clears, and it's the whole difference between the two techniques. The pivot is no longer a bystander that merely forces the wings; it's one of the places Z might land.
In exchange, the pattern is easier to find, because the pivot is a three-candidate cell rather than a pair, and there are far more of those in a hard grid.
Example
A worked example.
Find a three-candidate pivot
Look for a cell with exactly three candidates. Unlike the XY-Wing, the pivot here isn't a pair, so a grid full of three-mark cells is where to start.
Look for two wings in its units
In the cells that share a unit with the pivot, find two two-candidate cells, each holding a different pair drawn from the pivot's three digits.
Check the shared digit
Both wings must carry the same third digit Z, the one they have in common and the pivot also holds. If they don't, this isn't an XYZ-Wing.
Clear Z where all three are seen
Remove Z from every cell that can see the pivot and both wings. That's usually one or two cells, sometimes none, and the pattern earns nothing when there are none.
Spotting it
How to spot it.
- Start from three-candidate cells, not from pairs. The pivot is the unusual part of this pattern, so it's the cheapest thing to scan for.
- The wings must each see the pivot, and they may sit in any of its three units. Look in the row, the column and the box.
- Compare with XY-Wing: if the pivot turns out to have only two candidates after all, you have the simpler and stronger pattern, so use that.
- Before working the logic, check that some cell sees all three. An XYZ-Wing with no common observer proves something nobody can use.
Where it stops working
The elimination is narrow, so the technique often finds nothing worth having. It also needs the pivot to hold exactly three candidates: a fourth turns it into a different shape and the removal becomes wrong. As with every wing, it depends on a complete candidate list, because the pattern is made entirely of marks.
Keep going
Related topics.
New to sudoku? The rules take two minutes to read, and the first-puzzle walkthrough covers the basics that come before any of these.
Beyond the lessons.
The game's lessons stop at X-Wing, so there's no XYZ-Wing lesson. The eight it does teach will get you through almost any puzzle; this page is here for when you run into the pattern anyway.