XYZ-Wing
A three-candidate pivot and two wings share a removable digit.
An XYZ-Wing is a three-cell forcing pattern made of a pivot holding exactly three candidates and two "wing" cells, each of which shares two of those candidates with the pivot, with one single digit — the "z" — common to all three cells. In an XYZ-Wing sudoku position that shared digit is guaranteed to land on the pivot or one of its two wings no matter how the pivot resolves, so any cell that can see all three at once loses it as a candidate. It extends the XY-Wing by letting the pivot itself carry the eliminating digit rather than only the wings, which is exactly why the pattern is spelled with three letters instead of two.
Worked example
Here the pivot R1C1 holds {1,2,3}, wing R1C3 holds {1,3}, and wing R3C1 holds {2,3}; all three contain a 3 — the "z" digit — and both wings sit in box 1 alongside the pivot. If R1C1 is 1 then R1C3 is forced to 3; if R1C1 is 2 then R3C1 is forced to 3; and if R1C1 is 3 then the pivot is itself the 3 — so one of these three cells is always a 3. R3C3 shares box 1 with R1C1, column 3 with R1C3 and row 3 with R3C1, so it sees all three cells at once; a 3 there would collide with whichever cell is forced, and 3 is therefore eliminated from R3C3.
When to use it
Reach for the XYZ-Wing once basic singles, locked candidates and naked/hidden pairs have stalled and you are scanning a grid dotted with bivalue and trivalue cells for a short forcing chain. Spot it by picking a cell with exactly three candidates as the pivot, then looking in its own row, column and box for two bivalue cells whose pairs each overlap the pivot in two digits and share one digit in common. The elimination only exists where a single cell can see the pivot and both wings simultaneously, so the wings usually sit close to the pivot — often inside the same box, as in this box-1 example where R3C3 is the one cell that sees all three.
How to use it
- Find the pivot. Find a cell with exactly three candidates, x, y and z — the pivot of the wing.
- Find the two wings. Find two cells the pivot sees, one with candidates x and z, the other with y and z. Each is a subset of the pivot sharing z.
- Eliminate the shared digit. One of the pivot or wings must be z, so remove z from every cell that sees all three.
Frequently asked questions
What is an XYZ-Wing in Sudoku?
It is a three-cell elimination pattern: a pivot with three candidates {x,y,z}, one wing with {x,z} and another with {y,z}, all three cells sharing the digit z. Because the pivot must resolve to x, y or z, a z is guaranteed to appear in the pivot or one of the two wings, so z can be removed from any cell that sees all three at once. In the worked board that shared digit is 3, the pivot is R1C1={1,2,3} with wings R1C3={1,3} and R3C1={2,3}, and the 3 comes off R3C3.
How do I spot an XYZ-Wing on the grid?
Start from a trivalue cell as the pivot, then search its row, column and box for two bivalue cells that each share two candidates with the pivot and share one candidate with each other. Check that all three cells hold the same 'z' digit, then look for any single cell that sees the pivot and both wings — that intersection is where z is eliminated. In the example R1C1={1,2,3} is the pivot, R1C3={1,3} and R3C1={2,3} are the wings all carrying 3, and R3C3 is the only cell seeing all three.
What is the difference between an XYZ-Wing and an XY-Wing?
In an XY-Wing the pivot is bivalue and does not itself contain the shared digit z, so only the two wings can carry it and the target cell only needs to see those two wings. An XYZ-Wing has a trivalue pivot that also contains z, making the pivot a third possible home for that digit, so the eliminating cell must see all three cells — pivot and both wings — rather than just two. The extra 'Z' in the name is literally the pivot joining the wings in holding the digit z, as R1C1 does with the 3 in this board.