ALS-XZ

Two almost-locked sets linked by a restricted candidate lock a digit.

An Almost-Locked Set (ALS) is a run of N cells that together hold exactly N+1 candidate digits — strip away any one of those digits and the remaining cells snap into a locked set. The ALS-XZ sudoku technique bridges two such sets that share two digits: a "restricted common" X, whose candidates in both sets all see one another so X can occupy only one of the sets, and a second shared digit Z. Because X is denied to whichever set misses out, that set locks and is forced to place Z, pinning Z inside the two sets — so you can strike Z from any outside cell that sees every Z in both.

Worked example

Two almost-locked sets in row 1 — R1C1/R1C2={1,2,3} and R1C5/R1C6={1,2,6} — share restricted common 1 and common 2. The set left without the 1 must place 2, so R1C9 loses its 2.123123126126279
Two almost-locked sets in row 1 — R1C1/R1C2={1,2,3} and R1C5/R1C6={1,2,6} — share restricted common 1 and common 2. The set left without the 1 must place 2, so R1C9 loses its 2.

Row 1 carries two almost-locked sets. Set A is R1C1 and R1C2, both {1,2,3}, and Set B is R1C5 and R1C6, both {1,2,6} — two cells over three digits apiece. Their restricted common is 1, since every 1 across all four cells lies in row 1, so only one set can actually take it. Whichever set is denied the 1 becomes a locked pair and is forced onto its 2, which means a 2 must appear somewhere among these four cells. R1C9={2,7,9} sees all of them, so it can never be that 2 — the 2 is eliminated from R1C9.

When to use it

Save ALS-XZ for the late game — reach for it only once singles, subsets, fish and wings are spent and the grid is thick with three-candidate cells. Scan for two almost-locked sets (N cells, N+1 digits) that share two digits, then test whether one shared digit is "restricted": all its occurrences across both sets must see each other. If so, the other shared digit falls from any cell that sees it in both sets — exactly as the 2 drops from R1C9 above.

How to use it

  1. Find two almost-locked sets. An almost-locked set is a group of cells in one unit with exactly one more candidate than cells. Find two of them that share two candidates, X and Z.
  2. Check the restricted common X. X must be restricted: every cell holding X across both sets must see the others, so X can land in at most one of the two sets.
  3. Lock and eliminate Z. The set that misses X collapses to a locked set and must place Z, so Z is trapped in the two sets — remove Z from any outside cell that sees every Z in both sets.
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Frequently asked questions

What is an ALS-XZ in Sudoku?

ALS-XZ links two Almost-Locked Sets that share two digits. One shared digit, X, is the restricted common — its cells in both sets all see each other, so it can be placed in only one of the two sets — while the other shared digit, Z, becomes confined to the pair of sets. Any outside cell that sees every Z in both sets can then have Z removed, the way R1C9 loses its 2 in the worked example above.

How do I spot an ALS-XZ on the grid?

First work down to a dense late-game grid, then look for two groups of N cells that each hold exactly N+1 candidates and share two digits. Confirm that one shared digit is restricted — every occurrence of it across the two sets must be mutually visible. In the example Set A (R1C1, R1C2 = {1,2,3}) and Set B (R1C5, R1C6 = {1,2,6}) share 1 and 2, and the 1 is restricted because all four cells sit in row 1.

How is ALS-XZ different from an XY-Wing?

An XY-Wing chains three bivalue cells — cells with exactly two candidates — pivoting one against two pincers. ALS-XZ generalises that same idea to multi-cell sets: each end can be an Almost-Locked Set of any size, and a lone bivalue cell is simply the smallest possible ALS. So an XY-Wing is really a degenerate, special case of the broader ALS-XZ pattern.