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.
ALS-XZ is the foundational technique in the Almost Locked Set family. An Almost Locked Set is a group of N cells in a unit (or visible from each other) holding exactly N+1 different candidates. If you could remove one candidate from the set, the remaining N candidates would have to fill exactly the N cells — a locked set. The "almost" comes from the extra candidate that's preventing the lock.
The rule
Take two Almost Locked Sets A and B that don't overlap. Suppose they share two common candidates — call them x and z. Suppose further that every cell holding x in A sees every cell holding x in B (a restricted common on x). Then z can be eliminated from any cell outside both sets that sees every instance of z in A and every instance of z in B.
The argument: if A doesn't contain z, then A is locked on its non-z candidates. That forces x out of A, which in turn (via the restricted-common condition) forces x to be in B, locking B on x. Now B contains x, so B doesn't contain z either. Contradiction averted by the elimination — z must occupy a cell outside both ALSes that sees neither, which means cells that see all z-bearing cells of either ALS can't be z.
The same argument runs with A and B swapped, giving the symmetric elimination: x can be eliminated from cells that see every instance of x in both sets, provided z is the restricted common in the symmetric direction.
Why ALSes underlie a lot of advanced reasoning
Almost Locked Sets generalise the wing patterns. Y-wing, XYZ-wing, and WXYZ-wing are all ALS-XZ patterns where one of the two ALSes is a single bivalue cell. Many AIC chains pass through ALSes as if they were single steps. A solver fluent in ALS reasoning can replace several wing techniques and chain segments with one unified argument.
The cost is conceptual: the ALS framework asks the solver to identify groups of cells with N+1 candidates in N cells, which is harder than spotting a bivalue. The reward is that ALSes appear all over hard and expert puzzles, often where no named pattern would have caught the elimination.
Spotting ALSes
A useful starting point: every bivalue cell is an ALS (one cell, two candidates). Every naked pair is an ALS (two cells, three candidates if you include a third candidate that's already excluded). Naked triples and quads with one extra candidate are the larger ALSes. Once you start counting "cells in this group" versus "candidates among them," ALSes surface across the grid.
ALS-XZ is the simplest interaction; ALS-XY-wing and longer chains of ALSes generalise the pattern further.
See also
- ALS-XY-wing— Three Almost Locked Sets in a Y-wing-like configuration. Generalises ALS-XZ to a longer chain and surfaces eliminations that a single-pair ALS interaction would miss.
- 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.
- 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.
- WXYZ-wing— A four-cell wing pattern. Three pivot cells share a fourth candidate that all see the wing cell, eliminating that fourth candidate from any cell that sees all four.
- Candidate— A digit (1–9) a cell could still legally hold — one not yet ruled out by anything in its row, column, or 3×3 box. Every empty cell has between one and nine.
Read more
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