sudokly
Advanced technique

XY-Wing

An XY-wing is a three-cell pattern. The pivot cell has candidates {X, Y}. Two wing cells see the pivot, with one wing holding {X, Z} and the other holding {Y, Z}. Whatever digit the pivot takes, one of the wings must hold Z, so Z can be eliminated from any cell that sees both wings.

XY-wing is the first true chain technique. It tracks what happens across three cells linked by a shared digit, and the conclusion is forced regardless of which value any single cell takes.

How it works

Find a bivalue cell. Call its two candidates X and Y. This is the pivot. Now find a second bivalue cell that sees the pivot (shares a row, column or box) and holds candidates {X, Z}. Then find a third bivalue cell that also sees the pivot and holds {Y, Z}. Z is the shared digit. The pivot does not contain Z directly.

Now follow the logic. If the pivot takes X, the {X, Z} wing must take Z. If the pivot takes Y, the {Y, Z} wing must take Z. Either way one of the two wings ends up with Z. So any cell that sees both wings cannot contain Z.

X YX ZY Z¬ZPIVOT SEES BOTH WINGS. TARGET SEES BOTH WINGS
Pivot {X, Y} sees both wings. Wherever the pivot lands, one of the two wings becomes Z. Eliminate Z from cells seeing both wings.

When to look for it

After X-wing and fish techniques fail. The grid needs lots of bivalue cells. Late-game expert and evil puzzles, where candidate sets have collapsed, are the natural habitat.

Start by listing every bivalue cell. Try each one as a pivot in turn. The search is finite and almost always finds at least one wing within a few seconds.

Step-by-step example

  1. Pick a bivalue cell with candidates {X, Y}. Call it the pivot.
  2. List bivalue cells that see the pivot. From those, find one containing X plus a third digit Z. Call it the X-wing.
  3. From the same list, find another bivalue cell containing Y plus the same Z. Call it the Y-wing.
  4. Confirm both wings see the pivot. They do not need to see each other.
  5. Find cells that see both wings simultaneously. Erase Z from them.
  6. Re-scan. The Z eliminations often create a single or another chain opportunity.

Tips for spotting it

  • Find bivalue cells first. The chain is built from them.
  • The pivot must see both wings. Wings do not need to see each other.
  • Each of the three cells must be exactly bivalue. Three candidates anywhere kills the pattern.
  • Z is the elimination digit. Always.

Common mistakes

  • Wings with three candidates. The structure requires exactly two per cell.
  • Eliminating X or Y instead of Z.
  • Forgetting the visibility condition. The pivot must see both wings.
  • Confusing XY-wing with XYZ-wing, which has a trivalue pivot.

Practise it

Try expert sudoku. Mark every candidate fully and list the bivalue cells before doing anything else. Test each one as a pivot. After a few solves, XY-wing becomes the first technique you reach for when X-wing fails. Then graduate to XYZ-wing and coloring.

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