ALS Chains in Sudoku: Linking Almost Locked Sets Step by Step

ALS chains in Sudoku turn sets into chain nodes

ALS chains in Sudoku extend the idea of an Almost Locked Set beyond a pair of interacting groups. Each ALS contains N cells and N plus 1 candidates inside one row, column, or box. Neighboring ALSes in the chain are connected by restricted common candidates, or RCCs. An RCC is shared by two sets in such a way that the digit cannot appear in both sets at once because all relevant occurrences in one set see all relevant occurrences in the other. This restricted link lets a missing candidate in one ALS force the next ALS toward a different candidate, creating a sequence of implications. The first and last ALS must share an endpoint candidate for the usual elimination. ALS chains in Sudoku may involve only a few cells or quite large sets, but the principle remains a chain of nearly locked groups rather than a chain of individual bivalue cells.

Read ALS chains in Sudoku from one endpoint to the other

Imagine the first ALS and the last ALS both contain candidate Z. If Z is absent from the first ALS, that set becomes locked around its remaining candidates and forces the first RCC relationship. That restriction pushes the next ALS, which pushes the next, until the chain guarantees Z in the final ALS. If Z is present in the first ALS, then Z is already secured at the starting end. In either case, Z must occur in at least one endpoint ALS. A cell outside the chain that sees every possible Z in both endpoint sets cannot contain Z. That is the elimination. Adjacent RCCs must be chosen carefully, and the same restricted candidate cannot simply be repeated in a way that breaks the implication sequence. For hand solving, use short ALS chains first and write the RCC between each pair so the logic can be checked link by link.

Build ALS chains in Sudoku from visible small ALSes

A practical search for ALS chains in Sudoku starts with bivalue cells and two-cell sets containing three candidates. Mark one ALS, then look for a nearby set that shares an RCC. Continue only if the next set offers a different useful RCC toward another region. Once a short chain forms, compare the candidate lists of the first and last ALS for a common endpoint digit. Finally, search for targets that see all instances of that digit in both ends. Do not collect sets merely because they overlap visually; every connection needs the restricted-common condition. Shapedoku players can practice this with shape candidates on Extreme boards, where an ALS might contain two cells and three shapes rather than three digits. The structure is identical. ALS chains in Sudoku are demanding mainly because of bookkeeping, so clean Notes and compact notation matter more than speed. When a shorter XY-Chain, AIC, or ALS-XY-Wing proves the same result, use the simpler explanation.

When ALS chains in Sudoku grow beyond three or four sets, annotate each link with its RCC rather than relying on the candidate lists alone. A compact sequence such as A -4- B -7- C -2- D immediately shows whether adjacent links accidentally reuse the same restricted digit or whether one set lacks the candidate required for the next implication. Then write the common endpoint digit beside A and D. This notation turns a visually scattered pattern into a readable argument. If you cannot reconstruct why each RCC is restricted from the notes, shorten the chain or find a simpler explanation.

1. Find a small ALS and list its complete candidate set.
2. Connect it to a second ALS only through a valid restricted common candidate.
3. Continue with different useful RCCs rather than repeating links blindly.
4. Check whether the first and last ALS share an endpoint candidate.
5. Eliminate that endpoint candidate only from cells that see all endpoint occurrences.
6. Prefer short, auditable chains while you are learning the method.