Sudoku Y-wing (XY-wing): the bivalue trick that breaks expert puzzles

A bivalue cellis a cell that has exactly two possible candidates. No more, no less. On a hard or expert puzzle, you’ll find a handful of these once you fill in pencil marks. They’re the raw material for Y-wings.

The Y-wing pattern uses three bivalue cells whose candidate sets are arranged like a chain:

• One cell — call it the pivot — has candidates {A, B}.
• One wing has candidates {A, C}.
• The other wing has candidates {B, C}.

Each wing shares one candidate with the pivot and one candidate with the other wing. The shared “outer” letter between the two wings is C, which never appears in the pivot.

The other rule: each wing must “see” the pivot — meaning they share a row, column, or 3×3 box. They don’t need to see each other; that’s not required.

Here’s the magic. The pivot will hold either A or B — those are its only options.

• If the pivot is A: the {A,C} wing can’t also be A (because it sees the pivot via row/column/box), so that wing must be C.
• If the pivot is B: the {B,C} wing must be C for the same reason.

Either way, one of the two wings ends up as C. You don’t know which wing — but you know that any cell that sees both wings cannot be C, because one of them is C, and you can’t have two Cs in the same row, column, or box.

So: scan the cells that share a row, column, or 3×3 box with both wings (the intersection of their visibility). Erase C as a candidate from any such cell.

Suppose your pencil marks show:

The pivot at r1c1 has {2,7}. The wing at r1c5 shares the 2 (and adds 8). The wing at r4c1 shares the 7 (and also adds 8). The shared outer candidate C is 8.

Both wings “see” the pivot:

• r1c5 is in the same row as the pivot r1c1.
• r4c1 is in the same column as the pivot r1c1.

Now find cells that see BOTH wings (r1c5 and r4c1). The intersection of “in row 1 or column 5” (for r1c5) and “in row 4 or column 1” (for r4c1) gives you specific cells like r4c5 (which shares row 4 with r4c1 and column 5 with r1c5).

If r4c5 currently has 8 as a candidate, erase it. The Y-wing has done its work — and often, removing an 8 from one cell cascades into placements elsewhere.

Y-wings hide more than X-wings, but the search is mechanical:

1. List every bivalue cell. On an Expert puzzle there will be ten or twenty.
2. For each one, treat it as the candidate pivot. Note its two values {A, B}.
3. Find every bivalue cell that contains A (but not B) and sees this pivot. Each is a candidate {A, C} wing.
4. For each such wing, find every bivalue cell that contains B (but not A) AND contains the wing’s C value AND sees the pivot. That’s the matching {B, C} wing.
5. Once you have a pivot and two matching wings, find cells that see both wings and remove C from them.

Sounds tedious in prose; in practice you start seeing the shape after a dozen examples. The trick is having pencil marks complete — you can’t spot bivalue cells without them.

X-wingis for when scanning fails on a single digit and you can see two rows (or two columns) where the digit is restricted to exactly two cells in the same column pair (or row pair). It’s a rectangle, all on one digit.

Y-wing is for when scanning fails and you can spot three bivalue cells whose candidates form a triangle of {A,B}, {A,C}, {B,C}. It uses three digits and three cells.

On most Expert puzzles, one or the other will be available — sometimes both. If both look hard to spot, start with X-wing (easier to scan for: pick a digit, scan rows). If X-wing fails, fall through to Y-wing hunting.

The Y-wing is a 3-cell chain. The same idea extends to longer chains:

• XY-chain: an arbitrarily long chain of bivalue cells, each adjacent pair sharing one candidate. The Y-wing is the 3-cell special case.
• WXYZ-wing: a 4-cell variant where the pivot has three candidates instead of two.

These show up on Extreme puzzles. The Y-wing is usually the most you’ll need on Expert.