Techniques · Expert
W-wing
Two cells that never meet
A W-wing is two cells with the same two candidates. They do not see each other. A strong link on one of those digits joins them: some house holds that digit in exactly two cells, and each wing sees a different end of that link. At least one wing is the other digit. A cell that sees both wings loses it.
On this board the wings are row 2, column 2 and row 6, column 9. Both can be 4 or 5. Different rows, different columns, different boxes. Row 2, column 2 is in box 1. Row 6, column 9 is in box 6. Nothing in the pattern makes them look at each other. Draw the link between them and you get a lopsided W. I still have to count the cells before I trust the drawing.
The link is a 5 with two homes
The shared digit that does the joining is 5, not 4. Row 5 holds 5 in exactly two cells: row 5, column 2 and row 5, column 9. Both can be 5 or 7. The rest of row 5 is filled, so the 5 has nowhere else in the row to go. One of those two cells is 5.
Each wing sees one end and misses the other. Row 2, column 2 sees row 5, column 2 down column 2, and it does not see row 5, column 9. Row 6, column 9 sees row 5, column 9 up column 9, and it does not see row 5, column 2. That crossing is the wing. Column 2 is not the link. Its only 5s are the wing in row 2, column 2 and the link end in row 5, column 2. Row 2 still has more than two 5s. Row 2, column 3 lists 2, 4, 5, and 9, and row 2, column 7 lists 2, 3, 4, and 9.
One of the wings is 4
If row 5, column 2 is 5, then row 2, column 2 cannot be 5, so that wing is 4. If row 5, column 9 is 5, then row 6, column 9 cannot be 5, so that wing is 4. The row cannot put its 5 in both ends. One story or the other, a wing is 4. You do not learn which wing. A cell that sees both of them cannot be 4.
Row 5, column 2 and row 5, column 9 are also a naked pair, 5 and 7. The pair is idle. Every other cell in the row is already filled, so there is nothing for 5 or 7 to leave. The W-wing uses those two cells as a link, not as a cleanup.
The 4 that sees both wings
Row 2, column 2 and row 6, column 9 are the wings. Row 5 ties them with a 5. Step through. One 4 leaves, and that cell becomes 2.
See the pattern
Look at these two cells.
What the cells actually hold
Row 2, column 1 is 1. Row 2, column 4 is 6, row 2, column 5 is 8, and row 2, column 6 is 7. Row 2, column 2 can be 4 or 5. Row 6 is the other wing's row. Row 6, column 1 is 3, row 6, column 2 is 2, row 6, column 3 is 8, row 6, column 5 is 1, and row 6, column 6 is 6. Row 6, column 9 can be 4 or 5. Same two digits, and the cells do not see each other.
Row 2, column 9 can be 2 or 4. It shares row 2 with row 2, column 2, and it shares column 9 with row 6, column 9. Both wings. The 4 leaves. 2 is all that remains, so the step places 2 in row 2, column 9.
One wing is a near miss
Row 2, column 3 can be 2, 4, 5, or 9. It sees row 2, column 2, in the row and in box 1. It does not see row 6, column 9. The 4 stays. I have deleted that 4 because it sat next to a wing. One wing is not the pattern.
An XY-wing needs a pivot that sees both wings and carries a different pair for each of them. Nothing here is that pivot. Row 5, column 2 sees only one wing, and its candidates are 5 and 7, not a pair drawn from 4 and 5. A naked pair would need the two wings in one house. They are not in one house. If they could see each other, this would be a pair and I would take the pair.
Between the XYZ-wing and the fish
W-wings are Expert. They sit after the XYZ-wing and before the swordfish. A puzzle that needs one is not Mental. The 1,483 archived NYT puzzles never need a W-wing. Their hardest steps stop at triples.
I look for two matching two-digit cells that do not touch, then for a digit of theirs that has exactly two homes in a house, with each cell seeing one home. If the strong link is not there, the two cells are just two cells. The link is the whole trick.
That jumps a puzzle forward until W-wing is the next move. Or start a full Expert puzzle.