ALS Chains in Sudoku: Linking Almost-Locked Sets Into One Chain

Step 1: Understand the Basic Almost-Locked Set

An Almost-Locked Set (ALS) is a group of N cells that together contain exactly N+1 candidate digits. For example, three cells that collectively hold candidates for four different digits form an ALS. The key property is that at least one digit in the set must be false, because if all N+1 digits were true, they would need N+1 cells to hold them. This creates a strong link between the 'extra' candidates: at least one of them must be eliminated from the set. Think of an ALS as a nearly-complete unit where the internal candidates have a forced relationship. This internal logic becomes the building block for chains.

Step 2: Identify Restricted Common Candidates

Restricted common candidates are the glue that connects two ALSs in a chain. A digit is a restricted common if it appears as a candidate in both ALSs but is constrained so it cannot be true in both sets simultaneously. Typically, this happens when the shared candidate cells 'see' each other (are in the same row, column, or box). The critical rule is this: if the restricted common digit were true in the first ALS, it would be eliminated from the second ALS, and vice versa. This mutual exclusion creates a weak link between the two ALSs regarding that digit. When searching for an ALS chain, first find two promising ALSs and check for digits they share in mutually visible cells.

Step 3: Construct the Alternating Link Chain

An ALS chain connects sets by alternating strong links (within an ALS) and weak links (between ALSs via a restricted common). Start with one candidate in the first ALS. The strong link inside that ALS says: if this candidate is false, then a specific other candidate in the same ALS must be true. Then, use a restricted common candidate to create a weak link to the next ALS: if that 'true' candidate is a restricted common, then it cannot be true in the next set. This forces a specific candidate in the next ALS to be true. This alternating pattern continues. The chain effectively proves that if the starting candidate is false, the ending candidate must be true. This is the core logic of all Alternating Inference Chains.

Step 4: Apply the Elimination Rule

The power of the ALS chain lies in the elimination it enables. Once you have a valid chain connecting a starting candidate (A) to an ending candidate (Z), you have proven that A and Z cannot both be false. One of them must be true. Therefore, any cell that sees both a cell where A could be placed and a cell where Z could be placed cannot contain the digit they share. This shared digit is often, but not always, a restricted common from one of the links. You eliminate this digit from all cells that have visibility to both ends of the chain. This elimination can crack open a stalled puzzle, as seen in many advanced solve paths where basic methods like Strong and Weak Links alone are insufficient.

Step 5: Follow a Logical Search Order

To find ALS chains efficiently, follow a systematic search. First, scan the puzzle for promising ALSs—look for small groups of empty cells in a row, column, or box that share a limited pool of candidates. Next, for each ALS, list its candidate digits. Then, look for another ALS that shares one or more candidates with the first in a restricted way. Check if you can form a chain where the shared digit acts as a weak link. Finally, check the ends of the potential chain. Do they share a common digit? Can that digit be eliminated from any cells that see both end points? Practicing this search pattern helps internalize the process. The provided puzzle, after many singles, reaches a point where such a chain could be the key to progress, though its specific solution path uses only Naked Singles.

• Start by looking for ALSs in heavily constrained rows, columns, or boxes.
• Write down the full candidate list for each potential ALS to spot restricted commons easily.
• Remember the chain must alternate: strong link (within ALS), weak link (restricted common), strong link, etc.

Step 6: Differentiate from Simpler Techniques

It's crucial to distinguish an ALS chain from a single ALS elimination or a simpler XY-Chain. A single ALS can sometimes eliminate candidates that see all instances of a digit within the set, but it doesn't involve chaining multiple sets. An ALS chain explicitly connects multiple ALSs, multiplying their logical power. Compared to a basic XY-Chain, which links single cells, an ALS chain links groups of cells, making it a more generalized and often more discoverable pattern. This technique is a subset of AIC With Groups, where the groups are specifically Almost-Locked Sets. Understanding this hierarchy helps you know when to deploy this advanced tool.