Multi-digit chains use the same strong-link and weak-link language as single-digit chains, but the nodes are individual candidates rather than whole cells.

Conjugate pair strong links connect the same digit in two cells when those are the only two positions in a house. Bivalue cell strong links connect the two candidates inside one bivalue cell: if one is false, the other must be true. Weak links connect candidates that cannot both be true, either because they are the same digit in a shared house or different digits in the same cell.

This is what lets a chain switch digits while remaining logically valid.

Every cell bivalue. Consecutive cells share a house and one digit. If start and end digits match, at least one endpoint must contain that digit. Eliminate from cells seeing both endpoints.

A three-cell XY-Chain is an XY-Wing. Level 10 (Master).

An Alternating Inference Chain operates on the 729 possible candidate nodes: 81 cells times 9 digits. It alternates strong and weak links, and it can use links between candidates of the same digit, between the two candidates in a bivalue cell, or between candidates that conflict inside one cell.

Common useful outcomes include:

Type 1: The chain endpoints are the same digit in different cells. At least one endpoint must be true, so eliminate that digit from cells seeing both endpoints.

Type 2: The chain endpoints are different digits in the same cell. At least one endpoint must be true, so eliminate all other candidates from that cell.

Discontinuous loop: The chain closes with two links of the same type at one candidate. Two strong links meeting force that candidate true; two weak links meeting make that candidate false.

Level 12 (Extreme).

XY-Chain: only bivalue cells, digit switches happen inside those bivalue cells, and the common result is a Type 1 endpoint elimination. Level 10.

AIC: can use conjugate links, bivalue-cell links, grouped links, and other candidate conflicts. It can produce endpoint eliminations, cell reductions, placements, and discontinuous-loop eliminations. Level 12.

XY-Chain is a special case of AIC. AIC is strictly more powerful but much harder to find manually.

XY-Chains: Identify bivalue cells, build an adjacency map through shared candidates, and search for chains whose endpoints contain the same digit.

AICs: Start with reliable strong links such as conjugate pairs and bivalue cells. Extend the chain by alternating link types, and constantly check whether the endpoints produce an elimination or whether the chain closes into a useful discontinuity. Computer solvers often search the 729-node candidate graph directly.

Many simpler techniques are special cases of AIC or can be expressed naturally as AICs: XY-Wing, Skyscraper, W-Wing, Remote Pairs, and many coloring arguments. A solver implementing broad AIC search can reproduce many named techniques. However, the named techniques are faster to find, easier to explain, and better for human solving, so they are usually applied first.

XY-Chain (Level 10) threads through bivalue cells and is best understood as a restricted AIC. AIC (Level 12) operates on the full candidate graph and can switch digits, use many kinds of strong links, and produce several deduction types. Together they represent the culmination of chain-based solving.