3 Strong Links
3 Strong Links is the general case of the three-link strong-links family — three conjugate pairs on the same digit in any combination of houses where one cell from each neighbouring pair sees the other. The bridge cells can't both hold the digit, so at least one opposite endpoint must, eliminating the digit from any cell that sees both endpoints.
What are 3 Strong Links?
3 Strong Links is a single-digit pattern built from three conjugate pairs — three houses where a particular candidate appears in exactly two cells. The houses can be any mix of rows, columns, and blocks, provided one cell from each neighbouring pair shares a line with a cell from the next. Those two shared-line cells are the bridges: because they see each other, they can't both hold the digit. Since each conjugate pair must place the digit in one of its two cells, and the bridges can't supply both ends of the chain, at least one of the two outer endpoints — the roof — has to contain the digit. Any cell outside the pattern that shares a house with both of those endpoints can't also hold the digit, so it gets eliminated.
How to spot it
Like the other strong-links patterns, 3 Strong Links doesn't jump out the way singles or locked candidates do — you find it by scanning one digit at a time. Pick a candidate digit and look for houses where it's restricted to exactly two cells. When you find three such houses, check whether one cell from each neighbouring pair shares a row, column, or block with the next. If they do, you've got a 3 Strong Links chain: the shared-line cells are the bridges, and the two cells at the ends of the chain are the roof. The elimination lands on any cell that sees both roof cells. The specific sub-cases have their own names — when all three pairs sit in parallel rows or columns it's a 3 Skyscrapers, and when the houses alternate between rows and columns it's a 3-String Kite — but the remaining combinations, where at least one pair lives in a block or the houses don't follow a named shape, are all called 3 Strong Links. They turn up regularly in harder puzzles once the easier techniques have run their course.
The logic
Each conjugate pair is a house where the digit has exactly two candidate cells, so one of them must be the digit. In a 3 Strong Links chain, one candidate from each neighbouring pair sits on a shared line — the bridge. Because the two bridge cells see each other, at most one of them can be the digit. If the first bridge cell is the digit, the second pair's digit falls on its opposite endpoint, and the chain stops there. If the first bridge cell is not the digit, the first pair's digit falls on its other cell, which is also a bridge to the next pair — and the same reasoning carries forward. Tracing the chain link by link, every case forces at least one of the two outer roof cells to contain the digit, so any cell that sees both roof cells can't also hold it. The same reasoning holds regardless of which combination of houses the three pairs live in.