Weak link
A relationship between two cells where at most one can hold the digit. Looser than a strong link — both might be other digits — and the steady half of every chain technique.
A weak link is the looser of the two relationships chain reasoning is built from. Two cells in a unit form a weak link on a digit if at most one of them can hold that digit. They might both be something else; what's forbidden is them both being the same digit at the same time.
The classic weak link is the no-repeats rule itself. Any two cells sharing a unit form a weak link on every digit — they can't both be that digit, because the unit only has room for one. Most weak links are this trivial kind, and that's exactly what makes them useful in chains: weak links are everywhere a chain might want to step.
What weak links contribute to chains
Weak links don't force placements directly. Their job in a chain is to interrupt: if a step has just placed a digit by way of a strong link, the next step is usually a weak link to a cell that can't be the same digit. That weak link rules the digit out of the next cell, which sets up the following strong-link step on a different digit, and the chain continues.
Alternating Inference Chains are the canonical example. Each step alternates strong and weak links along a sequence of candidates. The strong links place; the weak links eliminate; the chain ends with a placement or elimination two or more steps removed from where it started.
How they differ from strong links
A strong link is exclusive-or — one of the two cells is the digit, exactly. A weak link is "not both true" — both can be false. In practice, strong links carry placement information forward; weak links carry elimination information forward. Neither alone runs a useful chain. The two cooperate, alternating, until the chain closes.
See also
- Strong link— A relationship between two cells in a unit where a digit must occupy exactly one of them — the basic primitive that hidden singles, X-wings, and chain reasoning all rest on.
- Bilocation— When a digit has exactly two candidate cells in a unit. The underlying configuration that strong links, X-wings, and chain techniques all reach for as their starting shape.
- Candidate— A digit (1–9) a cell could still legally hold — one not yet ruled out by anything in its row, column, or 3×3 box. Every empty cell has between one and nine.
- Alternating Inference Chain (AIC)— The general-purpose chain technique. Alternates strong and weak links along a sequence of candidates, eliminating a digit from any cell that sees both endpoints' candidates.
- XY-chain— A chain of bivalue cells linked by shared candidates. Eliminates a digit from any cell that sees both endpoints — the workhorse intermediate-to-advanced chain technique.
Read more
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