W-wing

Two bivalue cells linked by a strong link eliminate the other candidate

How to use it

Two remote twins and a link between them. Find two bivalue cells with the same pair {x, y} that do not see each other. Then find a strong link on x — a line where x has exactly two possible homes, one seeing each twin. If neither twin were y, both would be x, but the link delivers x exactly once and each link home kills the x of the twin it sees: contradiction. So one twin or the other is y, and every cell seeing both twins loses y.

Harvest equal bivalue marks first, then scan for the connecting strong link. The eliminated y is often the nudge that unsticks the grid.

Why it is sound

Two cells hold {x, y} and are linked by a strong link on x: a unit where x fits exactly two homes, one seeing each wing. In every solution x occupies one link home, which excludes x from the wing that home sees; that wing is then y. So one wing or the other is y in every solution, and a cell seeing both wings can never be y.

A worked example

Two cells share the digit pair {1, 2}: r1c2 and r4c3. They do not see each other, but column 1 holds a strong link on 1 — the digit 1 fits there exactly at r1c1 and r4c1, one seeing each wing. Suppose neither wing were 2: both would be 1. But the link delivers 1 exactly once, and whichever link cell takes it kills the 1 of the wing it sees — contradiction. So one wing or the other is 2, and every cell seeing both wings loses 2: r1c3 is struck. Find the twins first, then hunt the connecting strong link — one line, two homes, one per wing.