W-Wing Sudoku - Eliminating Candidates from 2 Cells Linked by Strong Links

W-Wing is constructed as follows. Two cells are both bi-value cells with the same {X, Y} candidates. These two cells need not share a row, column, or block - they can be in distant positions. A 'strong link' must exist between them: specifically, in some row, column, or block, the candidate positions for X (or Y) are limited to exactly 2 cells, and those 2 cells are each visible from one of the two bi-value cells via X (or Y). When this structure holds, the third cell that simultaneously sees both bi-value cells can have one digit (specifically, the digit not used in the strong link) eliminated. The W-shaped logical structure gives it the name W-Wing.

Understanding W-Wing requires the concept of 'strong link.' A strong link refers to the state where a specific digit X 'can fit in exactly two cells within a unit.' These two cells have an exclusive relationship: 'if one is X, the other isn't; if the other is X, one isn't.' This is the same concept as conjugate pairs used in <a href="/en/articles/sudoku-coloring-technique/">Coloring</a>. In contrast, a 'weak link' only states 'if one of two cells is X, the other isn't,' allowing both to not be X. The strong link asserts 'one of them must be X,' while the weak link asserts 'both cannot be X.' W-Wing is constructed only with strong links.

<a href="/en/articles/xy-wing-technique/">XY-Wing</a> and W-Wing both handle bi-value cells but differ fundamentally in structure. XY-Wing uses 3 cells (pivot + 2 pincers), each holding different candidate pairs {X,Y}, {X,Z}, {Y,Z}. The 3 cells directly share units. W-Wing uses 2 cells, both holding the same {X, Y}. The 2 cells don't need to directly share units - they're connected through the strong link as a 'relay logical relationship.' Practically, W-Wing can sometimes be easier to spot on a board (just looking for bi-value cell pairs), but verifying the strong link takes effort. Both deserve a place in the Expert+ solving repertoire as contrasting 'bi-value utilization techniques.'

Efficient W-Wing scanning follows these steps. Step 1: list all cells with 2 candidates on the board. Step 2: extract pairs of cells with the same {X, Y} candidates. If there are 4 cells with candidate {3, 8}, try pairs from those 4. Step 3: for the chosen 2 cells, verify a strong link on X (e.g., 3) exists. Specifically, identify rows, columns, or blocks where '3' is visible from the chosen 2 cells, and check whether 3's candidates in those units are limited to 2 cells. Step 4: if the strong link is found, eliminate Y (e.g., 8) from any third cell that simultaneously sees the chosen 2 cells. Step 5: if X has no link, try the same procedure with Y. Step 6: if no matching pair, repeat with different candidate pairs. At Master/Extreme level, W-Wing can often be discovered before other techniques (XY-Wing, Swordfish).

The W-Wing requires you to discern a somewhat special structure - two identical two-candidate cells and a strong link connecting them - so at first it is easy to take time finding it. The knack for mastering it is to first focus on a pair of cells whose candidates are the same two numbers, and check whether one of those two numbers forms a strong link (candidates in only two cells) in some row, column, or block. The logic itself belongs to the same family as the XY-Wing, based on a disjunctive syllogism, and is sure reasoning containing no guesswork. When stuck on a hard puzzle of Expert level or above, it often becomes a breakthrough alongside the XY-Wing. If you confirm the structure one piece at a time without rushing, you can apply it reliably while avoiding wrong eliminations.