SuuudokuuuW-Wing Sudoku Technique
Updated
A W-Wing links two cells that hold the same pair of candidates through a strong link on one of those digits elsewhere on the board, which lets the other digit be erased from every cell both starting cells see.
When a W-Wing applies
A W-Wing starts from two cells, anywhere on the board, that both carry exactly the same two candidates, X and Y. Each cell must eventually hold one of the two, and since they carry the identical pair, exactly one of them holds X and the other holds Y — never both holding the same value, because if one held X, a strong link on X between them would force the other away from X and into Y.
The strong link is what makes the connection provable. Somewhere on the board, a unit — a row, column or box — has X confined to exactly two cells, one of which is one of the two W-Wing cells and the other of which sees the second W-Wing cell. That conjugate pair on X ties the two ends of the wing together: if the first cell is not X, the link forces X into its partner, which forces the second W-Wing cell away from X and into Y.
Once the two ends are shown to always split between X and Y, any cell that sees both of them cannot hold Y, since one of the two W-Wing cells always does.
Worked example
r8c1 and r9c5 both carry exactly 1 and 5. A strong link on 5 connects them, so the two cells always split the pair between them: whichever one ends up holding 5, the other is left holding 1. There is no way for both to hold 1 at once.
r8c6 sits in row 8 with r8c1 and in the same box as r9c5, so it sees both W-Wing cells. It was carrying 1 and 6 before the deduction. Since one of r8c1 or r9c5 always holds 1, r8c6 cannot hold 1 as well, and the solver removes it, leaving 6 behind as a naked single.
How to spot a W-Wing
- Find two matching bivalue cellsLook for two cells, anywhere on the board, that carry exactly the same two candidates and do not already see each other.
- Find a strong link on one of the two digitsCheck whether a row, column or box confines one of the two candidates to exactly two cells, with one end reaching each of the two starting cells.
- Confirm the link ties the two cells togetherThe strong link must connect a peer of the first bivalue cell to a peer of the second, so that the two cells are always forced apart between the two digits.
- Erase the other digit from shared peersRemove the non-linked digit from any cell that sees both of the original bivalue cells.
Common mistakes
- Using two bivalue cells that already see each other. A W-Wing’s value comes from tying together two cells that are not already peers.
- Treating any shared candidate as a strong link. The link must confine one digit to exactly two cells in some unit, not just have the digit present in both.
- Eliminating the linked digit instead of the other one. The strong-link digit stays in play at both ends; it is the remaining digit that gets erased from shared peers.
- Forgetting to check both digits as the possible link. A pair might only work as a W-Wing through one of its two candidates, not the other.
W-Wing FAQ
What is a strong link?
A strong link on a digit is a unit — a row, column or box — where that digit has exactly two remaining candidate cells. If one of the two cells is not the digit, the other one must be.
Do the two W-Wing cells need to be in the same row, column or box?
No. They can be anywhere on the board. What ties them together is the strong link elsewhere, not any direct relationship between the two cells themselves.
How is a W-Wing different from an XY-Wing?
An XY-Wing uses a shared pivot cell with two of the three values in play. A W-Wing has no pivot at all; instead, two identical bivalue cells are tied together by a strong link somewhere else on the board.
Is a W-Wing a kind of chain?
It is a short, two-step version of the same alternating strong-and-weak-link reasoning that the chain techniques later in this list use on a larger scale.