W-Wing Technique in Sudoku

The W-Wing Technique

The W-Wing appears to be a useful, and often quite useful, strategy to apply when solving Sudoku puzzles.

It deals with two pairs of cells that contain two candidates and are linked by a strong connection on one of those candidates. Once a strong connection is established, the other candidate can be removed from any cell that can "see" both of these bi-value cells.

The W-Wing operates as a chain of logical inference, but the proof is complicated to put down on paper. Fortunately, the geometry of the pattern itself is simple, especially in candidate filter mode.

Example 1

The two bivalue cells are r4c4 and r8c9, both of which are candidates for 5 and 9. They are linked by a strong connection on candidate 9 in Column 8—this means that this column is restricted to two possible placements of 9, such that if one is false, the other must be true.

One end of the strong link (r4c8 cube) can see (r4c4 cell) and the other (r8c8 cell) rs (r8c9 cell). Hence candidate 5 can be eliminated from every cell that r4c4 and r8c9 can "see".

Explanation:

It works because, in Column 8, one of the two 9s must be placed.

• If r4c8 equals 9, then r4c4 has to equal 5.
• If r8c8 equals 9, then r8c9 equals 5.

In both situations, r4c4 and r8c9 both will equal 5, therefore, any cell which has the potential to see both must not have the value 5.

Example 2

W-Wing, candidates 4 and 1:
Bivalue cells: r1c9 and r8c7 are connected by a strong link on candidate 1 in Column 3 (r1c3 and r8c3).

Candidate 4 can be removed from all cells that see both bivalue cells, namely r123c7 and r89c9.

Unlike XY-Wing and W-Wing, which focus on candidate-based logic, the X-Wing remains a classic pattern technique. Comparing them side by side gives you a well-rounded approach to Sudoku solving.