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: R2C4 has {7, 8}, a single bivalue cell (1 cell, 2 candidates). ALS B: R2C3 {4, 7, 8} and R3C3 {3, 8}, two cells sharing column 3 with 3 combined candidates {3, 4, 7, 8} (2 cells, 3 candidates = 2+1).
The restricted common candidate is X = 7. R2C4 has 7 and can see R2C3 (which also has 7) through row 2. The second common digit is Z = 8. If R2C4 = 7, then ALS B locks as {4, 8}/{3, 8} → must contain 8. If R2C4 = 8, then 8 is already placed. Either way, 8 is in one of the pattern cells.
R2C1 and R2C2 see all cells containing 8 in both sets (R2C4 via row 2, and R2C3 via row 2). Eliminate digit 8 from R2C1 and R2C2. R2C1 reduces to {6}.
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