Multi-Digit Chains

Strong and Weak Links in the Multi-Digit Context

Conjugate pair strong links (same digit, two cells). Bivalue cell strong links (same cell, two digits -- the engine for digit switching). Inter-cell weak links (same digit, shared house). Intra-cell weak links (same cell, different candidates in non-bivalue cells).

XY-Chain: Threading Through Bivalue Cells

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).

AIC: The Most General Chain Technique

Operates on the 729-node graph of all (cell, digit) pairs. Alternates strong and weak links freely. Type 1: Same digit, different cells -- eliminate from cells seeing both. Type 2: Different digits, same cell -- eliminate other candidates from cell. Type 3: Discontinuous loop -- eliminate the start digit from start cell. Level 12 (Extreme).

XY-Chain vs AIC

XY-Chain: only bivalue cell strong links, all cells bivalue, Type 1 only, Level 10. AIC: both link types, any cell, three deduction types, Level 12. XY-Chain is a special case of AIC. AIC is strictly more powerful but much harder to find manually.

How to Find XY-Chains and AICs

XY-Chains: Identify bivalue cells, build adjacency map, search for matching start/end digits. AICs: Start with known strong links, look for bridge cells, extend promising chains. Computer solvers use BFS/DFS on the 729-node graph.

Why AICs Are the Ultimate Technique

Many simpler techniques are special cases of AIC: XY-Wing, Skyscraper, W-Wing, Remote Pairs. A solver implementing only AICs could reproduce many named techniques. However, simpler techniques are faster to find, so they are applied first.

Summary

XY-Chain (Level 10) threads through bivalue cells. AIC (Level 12) operates on the full graph. Together they represent the culmination of chain-based solving.