Two Almost Locked Sets connected by a restricted common candidate. The other common candidate is eliminated from cells seeing both sets.
Also known as: Almost Locked Sets - XZ Rule
An Almost Locked Set (ALS) is a group of N cells in a single unit (row, column, or box) that collectively contain exactly N+1 candidates. If they had N candidates, they would be fully locked (like a Naked Pair, Triple, or Quad). The "almost" means they have one extra candidate, remove any one digit and the set locks.
ALS-XZ connects two Almost Locked Sets through a "restricted common candidate" (X). Digit X appears in both ALS A and ALS B, and all cells containing X in one ALS can see all cells containing X in the other. This means X can only be true in one of the two sets, if ALS A uses X, ALS B can't, and vice versa.
The second common digit (Z) also appears in both sets. Since one set must absorb X, the other set must lock without X, and Z must be part of that locked set. Therefore, any cell that can see all Z-candidates in both sets cannot contain Z. This is one of the most powerful elimination techniques in Sudoku.
First, find two Almost Locked Sets, groups of N cells with N+1 candidates each, where the cells share a single unit. A single bivalue cell counts as an ALS (1 cell, 2 candidates).
Check if the two ALS share two common digits. Call them X and Z. Verify that X is a "restricted common", every cell with X in ALS A can see every cell with X in ALS B. This ensures X can be true in at most one of the two sets.
Since X is restricted, it forces one ALS to lock (become a proper Locked Set) without X. That locked set will contain Z. The other ALS also contains Z. Any cell outside both sets that can see every cell containing Z in both ALS A and ALS B cannot be Z, one of those positions must hold Z.
Example 1: ALS-XZ
ALS A is the single bivalue cell R2C4 {2, 5} (1 cell, 2 candidates). ALS B is R5C4 {2, 5, 6}, R5C6 {2, 7}, R5C9 {6, 7} — three cells in row 5 holding four candidates {2, 5, 6, 7} (3 cells, 3+1 candidates).
The restricted common is X = 5: it appears in R2C4 (ALS A) and R5C4 (ALS B), and those two cells share column 4, so 5 can be true in at most one of the sets. That forces the other common digit, Z = 2, into one of the two ALS: if A takes 5 then B locks and must place 2; if A takes 2, done.
R6C4 sees every cell that holds 2 in both sets — R2C4 and R5C4 down column 4, and R5C6 in the same box. Since one of them must be 2, eliminate 2 from R6C4, which reduces to {5, 6}.
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