The W-Wing Technique in Sudoku

The shape of a W-Wing

A W-Wing uses two cells that hold exactly the same two candidates, call them A and B, and that do not see each other, meaning they share no row, column or box. Between those two cells runs a strong link on one of the values. In practice that means somewhere on the board B appears in only two cells of a single unit, a conjugate pair, and each of those two B cells lines up with one of your pair cells. The two matching pair cells and the connecting link together form the wing. The reason it works is clean. The strong link guarantees that one of its two ends is B. Whichever end turns out to be B forces the pair cell it can see to give up B, and since that pair cell holds only A and B, it must then become A. You do not know which of the two pair cells is pushed to A, but you know for certain that at least one of them is. That single guaranteed A is the whole engine of the technique, and everything after it is just finding the cells that engine rules out.

A worked example

Suppose r2c1 and r8c9 both hold only the pair three and seven, and they do not see each other, since they share no row, column or box. Now look at column four, where seven happens to be a candidate in just two cells, r2c4 and r8c4, a clean conjugate pair on seven. Notice that r2c4 shares row two with r2c1, while r8c4 shares row eight with r8c9. That column-four pair is exactly the strong link the technique needs, tying the two identical bivalue cells together into a W-Wing on the value seven. The shape can be surprisingly hard to see the first few times, because the three ingredients sit in different parts of the grid: two twin cells in opposite corners and a conjugate pair bridging them through a shared column. But once you train your eye to notice two cells with the same pair of candidates and then ask whether a strong link connects them on one of those values, the configuration starts to leap out on otherwise stuck boards.

The elimination

Follow the logic through to the elimination. One of r2c4 and r8c4 must be seven. If r2c4 is seven, then r2c1 cannot also be seven and so becomes three. If instead r8c4 is seven, then r8c9 becomes three. Either way, at least one of r2c1 and r8c9 ends up a three, no matter which way the link falls. That means any cell able to see both r2c1 and r8c9 cannot itself be three, because one of those two cells has already claimed the three within its line of sight. Here r2c9 sees r2c1 along row two and sees r8c9 down column nine, and r8c1 sees both cells as well, so you erase three from r2c9 and r8c1. Notice that the value you remove, three, is the pair value that was not used for the strong link; the linked value, seven, does the forcing, and the other value is the one that gets cleared from the cells in view of both ends.

How to hunt for one

1. Find two cells with the identical two candidates that do not share any unit.
2. Choose one of the two values and look for a unit where that value has exactly two candidate cells, a conjugate pair.
3. Check that those two conjugate cells each see one of your pair cells, one apiece.
4. If they do, erase the other value from every cell that sees both pair cells.

W-Wing versus XY-Wing

Beginners often confuse the W-Wing with the XY-Wing, so it is worth drawing the line clearly. An XY-Wing uses three different bivalue cells whose candidates form a small chain, pivoting on one of them, and all three must sit within seeing distance of each other. A W-Wing uses only two bivalue cells, but they have to be identical, and it borrows a separate conjugate pair elsewhere to bridge the gap between them. That is what lets the W-Wing reach right across the board: its two ends need never touch a common cell, because the strong link does the reaching for it. In your wider toolkit, the W-Wing earns eliminations that basic scanning and naked pairs simply cannot see, and its outline becomes obvious once you have spotted a handful. It sits comfortably beside the XY-Wing, and a great many boards that feel stuck will yield to one wing or the other, which is why it pays to learn both together. In Shapedoku the matching pairs read as two cells showing the same two glowing shapes, so twin bivalue cells are pleasantly easy to pick out as you scan.