What is the XY-Wing?
The XY-Wing is a medium-level candidate-elimination technique that uses three bivalue cells (cells with exactly two candidates) to remove a candidate from other cells. You’ll often see it in pencil-mark-heavy puzzles where simple singles, pairs, and triples aren’t enough to make progress. The method relies on one cell acting as a pivot (the XY cell) and two cells acting as pincers (the XZ and YZ cells). When the pattern is present, a specific candidate (Z) can be eliminated from any cell that sees both pincers.
How the pattern works (plain logic)
Label the pivot’s candidates as X and Y. The two pincers must be bivalue too: one with candidates X and Z, the other with Y and Z. The pivot must share a house (row, column, or box) with each pincer so that a choice in the pivot forces a choice in a pincer:
- If the pivot is X, then the pincer holding X and Z must be Z.
- If the pivot is Y, then the pincer holding Y and Z must be Z.
Either way, the pincers cannot both allow something other than Z: in each scenario at least one pincer becomes Z. Therefore any cell that can see both pincers cannot be Z, because in all cases at least one pincer would block Z in those houses.
When to look for an XY-Wing
- After you’ve reduced candidates with basic elimination (cross-hatching, naked singles, hidden pairs, etc.).
- In puzzles where many cells are still bivalue or have three candidates—XY-Wing requires three bivalue cells but the target to eliminate can have more.
- Focus on bivalue cells that can reach two distinct bivalue cells by house connections. A pivot that sees many cells is a good place to search.
Step-by-step example (follow the candidate-grid)
Below is a simple instructional layout. The pivot is at row 4 column 4 with candidates {1,2}. One pincer is at r4c7 with {2,3} and the other at r6c4 with {1,3}. The target cell that sees both pincers is at r6c7 and currently contains candidate 3 among others.
- Pivot r4c4 = {1,2} (call these X=1 and Y=2).
- Pincer A r4c7 = {2,3} (Y,Z) so here Z=3.
- Pincer B r6c4 = {1,3} (X,Z).
Check the logical cases:
- If r4c4 = 1 (X), then r6c4 cannot be 1 and must be 3 (Z).
- If r4c4 = 2 (Y), then r4c7 cannot be 2 and must be 3 (Z).
In either case at least one of the pincers is 3. Any cell that sees both pincers (for example r6c7, which shares row 6 with r6c4 and column 7 with r4c7) cannot be 3, because whichever pincer becomes 3 will prevent a 3 from appearing in r6c7 in that house. So we can safely eliminate candidate 3 from r6c7.
Why this is safe: you aren’t guessing a value for the pivot. You are using the pivot’s two possibilities to show the same forced result (a pincer = Z) in both cases, and that forces elimination of Z in cells that are in sight of both pincers.
Practical tips for spotting XY-Wings
- Work from bivalue cells. List all bivalue cells and consider each as a potential pivot. For each pivot XY, look for one bivalue cell with X and Z and another with Y and Z that are both seen by some common target cell.
- Use a scanning pattern: pick a pivot, then examine its row, column, and box for bivalue candidates that share exactly one candidate with the pivot.
- The pincers do NOT need to see each other. They must each see the pivot, and there must be at least one cell that sees both pincers. That target cell is where Z can be eliminated.
- Don’t spend too long searching. If you’re stuck for more than a minute, move on and return later with fresh eyes. As you practice, finding pivots and matching pincers becomes faster.
Common mistakes to avoid
- Trying to use non-bivalue cells as pivot or pincers. The pivot and both pincers must each be exactly bivalue.
- Requiring the pincers to see each other—this is not necessary.
- Eliminating Z from cells that don’t actually see both pincers.
Practice and progression
Start by marking bivalue cells in a candidate grid or using software that shows pencil marks. Try to find at least one XY-Wing per puzzle when no simpler moves remain. With practice, the xy wing sudoku technique will become a reliable way to break through stubborn sections of a puzzle.
Example reference (see candidate-grid): the pivot r4c4 {1,2}, pincers r4c7 {2,3} and r6c4 {1,3}, allow elimination of 3 from r6c7. Try recreating similar arrangements in your own puzzles to build confidence.