Sudoku technique
XY-Wing
A pivot cell with two candidates {A, B} sees two wings: one holds {A, C}, the other {B, C}. Whichever value the pivot ends up taking, one of the wings takes C. Any cell seen by both wings at once 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 | 2 | 9 | ||||||
| 3 | 7 | |||||||
| 6 | 5 | 8 | ||||||
| 2 | 8 | 7 | 3 | |||||
| 8 | 9 | 2 | ||||||
| 6 | 7 | 5 | 8 | |||||
| 6 | 2 | 3 | ||||||
| 4 | 1 | |||||||
| 7 | 4 | 6 |
Demonstration
State of the grid right when the technique applies, after 19 earlier deductions.
Demo — watch first, you'll play right after.
- Technique's pattern
- Candidates removed
- Solved earlier
- Given (clue)
| 5 | 4 | 2 | 6 | 6 | 9 | 7 | 3 | |
| 3 | 8 | 2 | 7 | 5 | 6 | |||
| 7 | 6 | 5 | 2 | 8 | ||||
| 2 | 8 | 7 | 3 | 5 | ||||
| 8 | 9 | 7 | 2 | |||||
| 6 | 7 | 5 | 2 | 9 | 8 | |||
| 6 | 4 | 2 | 1 | 3 | 5 | |||
| 2 | 4 | 8 | 1 | |||||
| 7 | 5 | 4 | 2 | 6 |
XY-Wing
La case pivot (entourée) vaut 1 ou 4. Dans les deux cas, une des deux ailes prend le 6. Toute case vue par les deux ailes perd donc son 6.
Candidates removed:
- the 6 from cell L1C4
- the 6 from cell L1C5
- the 6 from cell L2C8
The XY-Wing involves three cells that each have only two candidates. A pivot cell and two pincers share three digits, so that one same digit is forced in one of the two pincers, whatever the pivot takes. Any cell that sees both pincers loses that digit.
The principle
The pattern has three cells with two candidates each. The pivot has candidates X and Y. The first pincer has X and Z, the second Y and Z. The pivot must see both pincers: each shares a row, a column or a box with it. That is what “seeing” means: two cells that see each other cannot hold the same digit. The pincers themselves do not need to see each other.
The reasoning goes through the two cases of the pivot. If the pivot is X, the XZ pincer, which sees it, can no longer be X: it is Z. If the pivot is Y, the YZ pincer can no longer be Y: it is Z. You do not know which case is the right one, but in both, one of the pincers is Z.
Any cell that sees both pincers at once therefore cannot hold Z: if it did, it would clash with the pincer that is Z, which you cannot point out. Remove Z from its candidates.
How to spot it
You need candidate notes. Start by finding the cells with only two candidates. Take one as the pivot, say {2, 5}, and look among the cells it sees for two other cells with two candidates: one that holds the 2 and a third digit, the other that holds the 5 and that same third digit. That third digit is Z.
Once you have the three cells, look for the cells that see both pincers. Remove Z from their notes if they hold it.
A worked example
The pivot, in row 1 column 1, has candidates 2 and 5. 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 can no longer be 2: it is 8. If the pivot is 5, the second pincer is 8. In every case, one of the pincers is an 8.
The cell in row 3 column 5 sees the first pincer through its column and the second through its row. It cannot be an 8. If it had candidates 3 and 8, remove the 8: 3 remains, and you can place it. The pivot does not need to see that cell.
Common mistakes
The most common mistake is to eliminate in the cells the pivot sees. The cells seen by both pincers are the ones that lose Z.
Another trap is to check that one pincer sees the pivot but forget the other. The pivot must see both pincers, each through a row, a column or a box. All three cells must also have only two candidates: a cell with three does not fit.
When to use it
The XY-Wing is expected at the Extreme level, along with the XYZ-Wing, the Swordfish and the Jellyfish. You do not know which value the pivot takes, but the elimination holds in both cases.
Frequently asked questions
What does it mean to “see” a cell?
Two cells see each other when they share a row, a column or a box. Two cells that see each other cannot hold the same digit.
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 both pincers at once.
What is the difference with the XYZ-Wing?
In the XY-Wing, the pivot has only two candidates, X and Y. In the XYZ-Wing, it has three, X, Y and Z, and Z is eliminated from the cells that see the pivot and both pincers at once.
Does an XY-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 XY-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.