XY-Wing Sudoku Technique

The first technique built on hypothetical reasoning: you do not learn what the pivot is, only that either way, one digit dies.

Also called: y wing, xy wing, bent triple.

What an XY-Wing is

Three cells, each with exactly two candidates. The pivot holds {X, Y}. One wing shares X and holds {X, Z}; the other shares Y and holds {Y, Z}. Both wings must be visible to the pivot — same row, column, or box.

The pivot is either X or Y. If it is X, the first wing is forced to Z. If it is Y, the second wing is forced to Z. One of the two wings is Z, always.

So any cell that can see both wings cannot be Z. It does not matter which wing wins.

The wings do not need to see each other — only the pivot. That bend is where the name "bent triple" comes from.

How to spot one

Collect every cell with exactly two candidates; nothing else can take part.
Treat each as a possible pivot and look along its row, column and box for two cells that share one of its digits each and a third digit with one another.
Then find the cells both wings see. If they are all empty of Z, the pattern is real but idle — move on rather than forcing it.

Worked example

r2c2 47 — Pivot
r2c8 79 — Wing, shares 7 along row 2
r6c2 49 — Wing, shares 4 down column 2
r6c8 19 — Sees both wings — the target

Cell references are row then column: r2c8 is row 2, column 8.

A pivot at r2c2 with wings at r2c8 and r6c2. The pivot sees the first wing along row 2 and the second down column 2.

Suppose the pivot is 4. Then the column-2 wing cannot be 4, so it is 9.
Suppose instead the pivot is 7. Then the row-2 wing cannot be 7, so it is 9.
Either way one of the two wings is a 9. The target cell shares row 6 with one wing and column 8 with the other, so it sees whichever wing turned out to be 9.
Strike 9 from r6c8, leaving it a naked single on 1.

What it eliminates

The shared digit Z, from every cell that can see both wings.

When you need it

On Extreme grids, usually as the alternative when no X-Wing or Swordfish is available. Many solvers find it easier to spot than a Swordfish despite sitting at the same level.

Difficulty here is set by how many clues are left in the grid: Easy starts from 46, Medium 40 and Hard 34, with Extreme and Insane stripped further still. Fewer clues is what makes the techniques below necessary — the fewer digits you start with, the deeper you have to reason.

Play a puzzle All solving techniques How to play Sudoku

Related techniques

X-Wing Sudoku Technique — The step where Sudoku stops being local. Everything before this works inside one unit; the X-Wing needs four cells in four different ones.
Swordfish Sudoku Technique — An X-Wing across three lines instead of two, with one wrinkle that makes it far harder to see.
Forcing Chains Sudoku Technique — The deepest reasoning in ordinary Sudoku, and the point at which the puzzle stops rewarding pattern recognition and starts rewarding bookkeeping.

Previous: Swordfish · Next: Chains and forcing chains