XYZ-Wing
An XYZ-Wing is a Y-Wing whose pivot has three candidates. The pivot has X, Y and Z; one wing sees the pivot and has X and Z, the other sees the pivot and has Y and Z. One of the three cells must be Z, so Z can be removed from every cell that sees all three. Because the pivot itself may be Z, the eliminated cell must see the pivot as well as both wings, which limits it to the box or line the cells share. It is the hardest technique Bare Sudoku uses.
When to use it
Only when everything else fails. XYZ-Wings are the rarest pattern in Bare Sudoku and appear only in Expert puzzles. Look for them when a three-candidate cell sits in a box with a two-candidate cell that shares two of its digits.
How to spot it
Start from cells with exactly three candidates; each is a possible pivot. Among its peers, look for two cells with two candidates each, both taken from the pivot's three, sharing exactly one digit Z. Then look for cells that see the pivot and both wings and contain Z. In practice one wing is in the pivot's box and the other in its row or column, and the eliminated cells are the remaining cells of that box in that row or column.
Example
| c1 | c2 | c3 | c4 | c5 | c6 | c7 | c8 | c9 | |
|---|---|---|---|---|---|---|---|---|---|
| r1 | 6 | 1 | 5 | 4 | 8 | 7 | 23 | 29 | 39 |
| r2 | 27 | 27 | 4 | 9 | 3 | 6 | 5 | 8 | 1 |
| r3 | 9 | 8 | 3 | 2 | 15 | 15 | 6 | 4 | 7 |
| r4 | 2578 | 4 | 9 | 1 | 256 | 258 | 278 | 3 | 568 |
| r5 | 12578 | 27 | 168 | 3 | 9 | 2458 | 278 | 26 | 4568 |
| r6 | 258 | 3 | 68 | 7 | 256 | 2458 | 9 | 1 | 4568 |
| r7 | 18 | 9 | 7 | 6 | 4 | 3 | 18 | 5 | 2 |
| r8 | 4 | 5 | 2 | 8 | 7 | 19 | 13 | 69 | 369 |
| r9 | 3 | 6 | 18 | 5 | 12 | 129 | 4 | 7 | 8 |
Bold digits are given, blue digits were placed while solving, small digits are candidates. Highlighted cells form the pattern, crossed-out candidates are removed, and a circled candidate is the digit to place.
- The pivot r8c9 has three candidates: 3, 6 and 9. It sees r1c9 with 3 and 9, and r8c8 with 6 and 9.
- If r8c9 is 9, done. If it is 3, r1c9 must be 9. If it is 6, r8c8 must be 9. In every case one of the three cells is 9.
- A cell that sees all three cannot be 9. Remove 9 from r9c9.
Common mistakes
- Eliminating from cells that see only the two wings. Unlike a Y-Wing, the pivot can be Z, so the cell must see the pivot too.
- Using wings whose candidates are not both contained in the pivot's three candidates.
- Confusing it with a Y-Wing: the difference is the third candidate in the pivot, and it changes which cells can be cleared.
In Bare Sudoku
Needed from level: Expert
In the game, the hint says: XYZ-Wing
Frequently asked questions
What is an XYZ-Wing in Sudoku?
A pivot with candidates X, Y and Z and two wings, one with X and Z and one with Y and Z, both seeing the pivot. One of the three cells is Z, so Z is removed from cells that see all three.
How is an XYZ-Wing different from a Y-Wing?
The pivot has three candidates instead of two, including Z. That makes the pivot a possible home for Z, so only cells that see the pivot and both wings can lose Z.
How often does an XYZ-Wing appear?
Rarely. In Bare Sudoku only Expert puzzles can need one, and most do not. Bare Sudoku checks it last, after X-Wing, Y-Wing and Swordfish.
Is there a WXYZ-Wing?
Yes, with a four-candidate pivot, but Bare Sudoku does not use it. Every puzzle in the game can be finished with the nine techniques of this guide.