Sudoku XYZ-wing: the trivalue pivot variant of Y-wing
The XYZ-wing extends the Y-wing pattern by allowing the pivot cell to have three candidates instead of two. The elimination logic stays similar but the “visibility” requirement gets stricter — a cell needs to see all three wing cells (pivot + both wings) for the elimination to apply. Less common than Y-wing, but real on Expert and Extreme puzzles.
The pattern
XYZ-wing has three cells sharing exactly three candidate digits, traditionally labelled x, y, z:
- Pivot: a cell with candidates
{x, y, z}(three candidates). - Wing 1: a cell with candidates
{x, z}that sees the pivot. - Wing 2: a cell with candidates
{y, z}that sees the pivot.
The shared digit between all three cells is z. The pivot has z, both wings have z. Wing 1 also has x (shared with pivot). Wing 2 also has y (shared with pivot).
The elimination
Whatever ends up at the pivot, it’s x, y, or z. Each scenario forces a wing to be z:
- Pivot = x → Wing 1 (which had
{x, z}) can’t be x → Wing 1 = z. - Pivot = y → Wing 2 (which had
{y, z}) can’t be y → Wing 2 = z. - Pivot = z → the pivot itself is z.
In every branch, one of the three cells (pivot or wing) is z. So any cell that sees all three of pivot, Wing 1, and Wing 2 cannot be z — because one of them already is. Eliminate z as a candidate from that cell.
The visibility constraint
This is the trickier part vs. plain Y-wing. The cell you eliminate z from must see all threeof pivot, Wing 1, and Wing 2. With Y-wing the elimination only had to see the two wings (because the pivot wasn’t a candidate for z).
In practice this means the elimination cell needs to share a row, column, or 3×3 box with each of the three cells. Often the only place this works is inside the pivot’s box, where one of the wings also sits. The pattern is most productive when:
- The pivot and one wing share a box.
- The other wing sits in the pivot’s row or column.
- The cells inside the pivot’s box (but outside the wing’s cell) can see the other wing through the shared row/column.
What it looks like on the grid
Vs. Y-wing (XY-wing)
- Y-wing: pivot is bivalue
{x, y}. Wings are{x, z}and{y, z}. Eliminate z from any cell seeing both wings. - XYZ-wing: pivot is trivalue
{x, y, z}. Same wings. Eliminate z from any cell seeing pivot and both wings.
The Y-wing is easier to apply because the pivot is eliminated from the visibility requirement. XYZ-wing is rarer because the pivot constrains your elimination cells more — but it’s still useful when you have a trivalue cell with the right candidate distribution.
For completeness, there’s also a WXYZ-wing(a quadrivalent pivot variant) and other longer chains, but they’re rare in published puzzles and mostly the domain of solvers building solver software.
When to scan for it
XYZ-wing is worth checking when:
- You’ve already tried Y-wings + the cheaper fish patterns.
- You have a trivalue cell with candidates
{x, y, z}. - You can spot bivalue cells with
{x, z}and{y, z}that both see the trivalue.
Scanning protocol: pick a trivalue cell, list its three candidates as x, y, z. For each candidate pair, look in the same row, column, or box for a bivalue cell with that pair plus the third candidate. If you find two such bivalues — one matching {x,z} and one matching {y,z} — check the cells that see all three.