W-Wing

The W-Wing is a pattern that uses two bi-value cells with the same pair of candidates and a strong link on one of those candidates to force eliminations. If the two bi-value cells do not see each other directly, but one candidate forms a strong link between them through a row, column, or box, you can often eliminate the other candidate from cells that see both bi-value cells.

When To Look For It

• Two cells are bi-value with the same pair of candidates, for example {a, b} in both cells.
• The two cells are in different houses so they do not see each other directly.
• There is a strong link on one candidate between the houses of the two cells. A strong link means that candidate occurs exactly twice in that house.
• There are target cells that can see both bi-value cells. These are the places where eliminations may happen.

How It Works

1. Find two bi-value cells with the same pair {a, b} that do not see each other.
2. Identify a strong link on candidate a that connects the houses of the two bi-value cells. This can be in a row, column, or box.
3. Because of the strong link, if one bi-value cell takes a, the other is forced to take b, and vice versa.
4. Any cell that can see both bi-value cells cannot contain b. Eliminate b from those target cells.

Quick Example

Suppose two cells each contain candidates {3, 7}. They are in different boxes and not in the same row or column. In one column, candidate 3 appears exactly twice and those two positions align with the rows of the two bi-value cells. The 3s form a strong link across that column. The setup means one {3, 7} cell must take 3 and the other must take 7. Any cell that sees both bi-value cells cannot be 7, so remove 7 from those cells.

Why It Works

The strong link on candidate a guarantees that only one end of the link can contain a. This forces the other bi-value cell to resolve to b. Since one of the paired cells must end up as b, any cell that sees both paired cells cannot take b without causing a contradiction.

Tips

• Mark bi-value cells first. A list of cells with exactly two candidates makes W-Wings much easier to spot.
• Hunt for strong links. Scan rows, columns, and boxes for candidates that appear exactly twice.
• Verify sight lines. Your eliminations are only valid for cells that see both bi-value cells.
• Convert progress. After eliminations, rescan for singles using Last Possible Number and Last Remaining Cell.

Common Mistakes To Avoid

• Using bi-value cells with different pairs. Both cells must share the same two candidates.
• Assuming a strong link without counting. Confirm that the linked candidate appears exactly twice in the house.
• Eliminating from cells that do not see both bi-value cells. The target must have a clear line of sight to each of the two cells.

Where You Will Use It

W-Wing is common in medium and hard puzzles. It is especially useful when simple singles have dried up but there are many bi-value cells and clear strong links in rows or columns.