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.
Alternating Inference Chains are the master chain technique. An AIC is a sequence of candidates connected by alternating strong links and weak links. Strong links propagate placements forward; weak links propagate eliminations forward. The chain starts at one candidate, alternates link types along its length, and ends at another candidate. Any cell that sees both end candidates can have the chain's "passing digit" eliminated.
The alternation rule
Each step in an AIC is a candidate of a particular digit in a particular cell — not a whole cell. The chain links connect candidates, not cells. Adjacent links must alternate between strong and weak. Two strong links in a row would skip the chain's reasoning; two weak links in a row would lose the case-by-case structure that makes the chain valid.
The starting and ending links must be strong. That's how the chain delivers its conclusion: the first strong link forces something; the last strong link forces something else; the alternations in between guarantee that whatever combination of true/false the chain takes, one of the two endpoints is true.
How AIC subsumes other chain techniques
XY-chain is the special case where every link is between two bivalue cells. Simple coloring is the special case where every link is on the same digit. Y-wing is the three-cell case of XY-chain. ALS-XZ is the AIC with Almost Locked Sets at the endpoints.
In other words, AIC is the framework; the named techniques are the patterns that show up often enough to deserve their own labels. A solver who fluently constructs AICs on demand can produce the equivalent of any of those techniques as a special case, plus eliminations that none of the named techniques cover.
Why most solvers learn it last
AIC is conceptually unified but operationally hard. Each step requires tracking two cases — the "true" branch and the "false" branch — at once, and the alternation rule has to hold strictly. Manual solvers often make notation mistakes that undermine the chain. The reward is that an AIC of length five or six can solve a stuck state on an extreme-difficulty puzzle in a single move; nothing simpler will reach that move.
A practical sequencing habit: don't reach for AIC before you've exhausted XY-chain and the named fish techniques. AIC's generality means it can solve more, but it also means more wrong attempts before the right chain surfaces.
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.
- 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.
- 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.
- Simple coloring— A technique that two-colours the strong-link graph of a single digit, then eliminates candidates that see both colours — the entry point into chain reasoning.
- ALS-XZ rule— An interaction between two Almost Locked Sets sharing a common candidate. Eliminates a second shared candidate from cells outside both sets that see all instances.
Read more
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