Sudoku technique
XYZ-Wing
An extension of the XY-Wing: the pivot has three candidates {A, B, C}, and its two wings hold {A, C} and {B, C}. Digit C has to belong to one of the three cells. Any cell seen by all three loses its candidate C.
A puzzle that requires it
Level Extreme — the puzzle is shown twice: right below, and again in the demo that follows.
| 5 | 4 | 1 | 3 | 7 | ||||
| 6 | 2 | 5 | 9 | |||||
| 9 | ||||||||
| 7 | 1 | 5 | ||||||
| 3 | 5 | |||||||
| 9 | 1 | 6 | ||||||
| 8 | ||||||||
| 8 | 3 | 6 | 4 | |||||
| 4 | 7 | 1 | 3 | 5 |
Demonstration
State of the grid right when the technique applies, after 14 earlier deductions.
Demo — watch first, you'll play right after.
- Technique's pattern
- Candidates removed
- Solved earlier
- Given (clue)
| 5 | 4 | 9 | 1 | 3 | 7 | |||
| 6 | 7 | 2 | 5 | 9 | ||||
| 9 | 7 | 5 | 8 | |||||
| 7 | 1 | 5 | 3 | |||||
| 3 | 2 | 1 | 7 | 5 | ||||
| 2 | 9 | 1 | 7 | 6 | ||||
| 7 | 8 | |||||||
| 8 | 3 | 6 | 5 | 7 | 4 | |||
| 4 | 7 | 1 | 8 | 3 | 5 |
XYZ-Wing
La case pivot (entourée) et ses deux ailes se partagent 1, 2, 8. L'un des trois prend forcément le 8 : toute case qui les voit tous les trois le perd.
Candidate removed:
- the 8 from cell L3C9
The XYZ-Wing extends the XY-Wing. The pivot has three candidates, X, Y and Z, and two pincers with two candidates each go with it: one with X and Z, the other with Y and Z. Digit Z has to be in one of the three cells. It is removed from the cells that see all three at once.
The principle
The pattern has three cells. The pivot has three candidates: X, Y and Z. The first pincer has two, X and Z. The second has two as well, Y and Z. The pivot must see both pincers, each through a row, a column or a box. That is what “seeing” means: two cells that see each other cannot hold the same digit.
The reasoning goes through the three possible values of the pivot. If it is X, the XZ pincer, which sees it, can no longer be X: it is Z. If it is Y, the YZ pincer is Z. And if it is Z, the pivot itself is Z. You do not know which case is the right one, but in all three, one of the three cells holds Z.
A cell that sees all three cells at once therefore cannot hold Z: it would clash with the one that does, which you cannot point out. Remove Z from its candidates.
How to spot it
You need candidate notes. First look for a cell with three candidates, say {2, 5, 8}. Among the cells it sees, look for two cells with two candidates: one that holds the 2 and the 8, the other that holds the 5 and the 8. Z is the digit common to both pincers, here the 8.
Then look for the cells that see all three cells, and remove Z from their notes.
A worked example
The pivot, in row 1 column 1, has candidates 2, 5 and 8. A first pincer, in row 1 column 5, has 2 and 8: it shares row 1 with the pivot. A second pincer, in row 3 column 2, has 5 and 8: it shares the top-left box with the pivot. The digit common to the pincers is 8.
If the pivot is 2, the first pincer is 8. If it is 5, the second pincer is 8. If it is 8, the pivot is. In every case, an 8 sits in one of the three cells.
The cell in row 1 column 2 sees the pivot, the first pincer through row 1 and the second through column 2. It cannot be an 8. If it had candidates 3 and 8, remove the 8: 3 remains, and you can place it.
Common mistakes
The most common mistake comes from the XY-Wing: removing Z from the cells that see only the two pincers. Here, the pivot holds Z too. In the example, the cell in row 3 column 5 sees both pincers but not the pivot: it keeps its 8, because the 8 may be in the pivot.
Another trap: the pincers must reuse the pivot’s digits, {X, Z} and {Y, Z} for a pivot {X, Y, Z}.
When to use it
The XYZ-Wing is expected at the Extreme level, along with the XY-Wing, the Swordfish and the Jellyfish. Look for it when the simpler techniques give nothing more. The XY-Wing covers the neighbouring case, where the pivot has only two candidates.
- Extreme sudoku
- Technique: the Swordfish
- Technique: the XY-Wing
- How to solve a sudoku
- Learn the techniques
Frequently asked questions
Why is Z only eliminated from cells that see all three cells?
Because Z can be in any of the three cells: the pivot, the first pincer or the second. To be sure of never clashing with it, a cell has to see all three.
What is the essential difference with the XY-Wing?
In the XY-Wing, the pivot has only two candidates, and Z is eliminated from the cells that see both pincers. In the XYZ-Wing, the pivot holds Z too: the cell must therefore see the pivot and both pincers.
Do the two pincers have to see each other?
No. Only the pivot has to see each of the two pincers. The cells that lose Z are those that see all three cells.
Does an XYZ-Wing place a digit?
No, it removes candidates. Those eliminations can then leave a Naked or Hidden Single, which lets you place a digit.
Practice
Every technique helps remove candidates ; once a cell has only one left, you place it.
Three puzzles where XYZ-Wing is the key move, served right before the step, with a coach.
Puzzle 1
Extreme
The technique is used once in this puzzle
Puzzle 2
Extreme
The technique is used once in this puzzle
Puzzle 3
Extreme
The technique is used twice in this puzzle
Puzzles change every day.