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XY-Wing Technique

1Definition: Also known simply as the Y-Wing technique, this involves three cells with three different candidates. But unlike a Triple, the cells are not all in the same Lane or Block, and each cell must contain exactly two candidates. They have a pattern in which the middle cell of the three has exactly one candidate in common with each of the other two (known as the "pincer" cells), and the pincer cells have exactly one candidate in common with each other.
Skeleton view of an XY-Wing. Pivot: C4; Pincers: C2 & E4. Pincers' houses overlap at E2.

XY-Wing in action. Pivot: B4.


2Look For: Cells with exactly two candidates, where three such cells line up to form a right angle (the "Y" of the wing -- though it would be more aptly named an L-Wing.)


3Consequence: Where the houses of the pincer cells overlap, the candidate they have in common can be erased.


4Why it Works: Think of the Pivot cell and either Pincer cell as half-matching pairs. The candidate that matches between these cells cannot be true for both cells: if it is true for one cell, then it must be false for the other, and since the other cell is a pair then it must solve to the candidate not held in common. Since the candidate not held in common with the Pivot is held in common between the pincer cells, we know at least one of the Pincer cells will solve to that candidate, and threfore it can be erased from the cell(s) where the Pincer cells houses overlap.


5In our example, if B4 solves to a 1, then B2 cannot be a 1, and threfore must be a 5.


6Conversely, if B4 should solve to a 4, then G4 cannot be a 4, and therefore must be a 5.


7So, whether B4 turns out to be a 1 or a 4, B2 or G4 will be a 5. [They could both solve to a 5, but that's not relevant here.]


8If a 5 appears in B2there cannot be a 5 anywhere else in B2's house.
If a 5 appears in G4 there cannot be a 5 anywhere else in G4's house.
So, even though we don't know which will be the case, we know that where B2's and G4's houses overlap a 5 cannot appear (since at least one of the houses covered by the overlap will already contain a 5.) Therefore, we can erase the candidate held in common by the pincers where their houses overlap: the 5 at G2.


9This causes a nice chain-reaction where G2 solves to an 8; D2 to a 1; B2 to a 5; B4 to a 1 (the only remaining 1 in column B); and D4 to an 8. But it stops there: G4 is not solved yet since B4 did not solve to a 4 (the matching candidate in this half-matching pair.)


10Now it's Your Turn!
Find the XY-Wing in this example, then erase the candidate from the cell where the pincer cell's houses overlap.

Hint


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