Y-Wing Chain

Y-Wing (sometimes called XY-Wing) is a short logical chain built from three cells that each have exactly two candidates. It's one of the most approachable "chain" techniques, and it shows up often enough in Expert puzzles to deserve a spot in any advanced solver's toolkit.

The Three Pieces

A Y-Wing uses three cells with three shared digits, call them A, B, and C:

• Pivot: a cell with candidates {A, B}.
• Wing 1: a cell with candidates {A, C}, sharing a unit with the pivot (same row, column, or box).
• Wing 2: a cell with candidates {B, C}, sharing a unit with the pivot (but not necessarily the same unit as Wing 1).

The pivot can see both wings. Each wing has one candidate in common with the pivot, and both wings share the third candidate C.

What the Pattern Tells You

Whatever the pivot turns out to be, one of the two wings must end up as C:

• If the pivot becomes A, then Wing 1 can't be A (it sees the pivot), so Wing 1 must be C.
• If the pivot becomes B, then Wing 2 can't be B (it also sees the pivot), so Wing 2 must be C.

Either way, one of the two wings is forced to be C. That means any cell that is visible to both wings — that shares a unit with each of them — cannot possibly be C. We don't know which wing will take C, but exactly one of them will, and that's enough to eliminate C from any cell that sees both.

Worked Example

Place the three cells on a grid:

• Pivot at r4c5 with candidates {2, 7}.
• Wing 1 at r4c8 with candidates {2, 9}. (Shares row 4 with the pivot.)
• Wing 2 at r6c5 with candidates {7, 9}. (Shares column 5 with the pivot.)

The common digit across the two wings is 9. If r4c5 is 2, then r4c8 can't be 2, so r4c8 is 9. If r4c5 is 7, then r6c5 can't be 7, so r6c5 is 9. Either way, one of r4c8 or r6c5 is 9.

Now look at any cell that sees both r4c8 and r6c5 — for example, r6c8 (same column as r4c8, same row as r6c5). Because one of r4c8 or r6c5 is guaranteed to be 9, r6c8 cannot be 9. You can delete 9 as a candidate from r6c8.

Any cell visible to both wings is a candidate for the same elimination. There may be several such cells, and the elimination applies to all of them.

How to Find a Y-Wing

1. Scan for "bivalue" cells — cells with exactly two candidates. These are the only cells that can participate.
2. Pick a candidate bivalue cell as the pivot.
3. Look at the cells it sees. From those, find two wing candidates whose candidate pairs together with the pivot's pair use exactly three digits in the {A, B}, {A, C}, {B, C} configuration.
4. Identify cells visible to both wings. The elimination target is the digit the two wings share — C — and the affected cells are any that sit in a common unit with both wings.
5. Remove C as a candidate from each of those cells.

Why It Works — in One Sentence

The pivot acts as a switch: whichever value it takes forces one of its neighbours to be C, so any cell that shares a unit with both neighbours cannot be C.

Common Mistakes

• Wings that don't see the pivot. Both wings must share a unit (row, column, or box) with the pivot. A wing in an unrelated part of the grid doesn't participate.
• Three-candidate cells. Each of the three cells must have exactly two candidates. A cell with three candidates can't fulfill the wing role cleanly.
• Wrong digit. The elimination targets the digit shared by the two wings (C in the examples here), not the digits shared with the pivot.
• Eliminating from a cell that only sees one wing. The target must see both wings, otherwise the chain doesn't reach it.

Y-Wing vs. XYZ-Wing

A close cousin is the XYZ-Wing. There the pivot has three candidates {A, B, C}, Wing 1 has {A, C}, and Wing 2 has {B, C}. The elimination target is still C, but now the target must see all three cells — pivot, Wing 1, and Wing 2 — because C is also a candidate in the pivot. XYZ-Wings are rarer and easier to miss, but they follow the same general logic.

When Y-Wing Helps

Y-Wing enters the picture when pencil marks start to have lots of bivalue cells scattered around the grid. Expert puzzles reach that state frequently, because earlier techniques reduce many cells down to two candidates without immediately forcing placements.

If a puzzle is stuck and several cells show exactly two candidates, walk through each of those cells as a potential pivot. Often, one of them will connect to two wings that share a common digit — and that digit can then be eliminated from one or more cells elsewhere in the grid.

Related Techniques

• X-Wing — a grid-spanning rectangle based on a single digit.
• Swordfish — the three-row extension of X-Wing.
• Locked Candidates — the box-line technique that usually precedes Y-Wing in a solve.

Practice

• Today's Expert puzzle

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