ALS-XZ Rule
The ALS-XZ rule is the doorway to Almost Locked Set logic — the family of techniques behind Sue de Coq and the Death Blossom. An almost locked set is a group of cells holding just one more candidate than it has cells: a bi-value cell is the smallest example, and any near-naked pair or triple qualifies. Link two of them cleverly and you can eliminate a digit neither could touch alone.
The idea
Take two almost locked sets, A and B, that share two digits, called X and Z. The digit X must be the restricted common: every cell holding X in A can see every cell holding X in B, so X can be true in only one of the two sets. That restriction locks the sets together — whichever one gives up X becomes fully locked. The consequence falls on the other shared digit, Z: no matter which way X resolves, one of the two sets is forced to contain Z. Therefore any cell outside both sets that can see every Z in A and every Z in B cannot itself be Z, and you remove it.
A worked example
Let set A be the bi-value cell r1c1 = {3,6} — one cell, two candidates, so almost locked. Let set B be two cells, r1c4 = {3,9} and r2c4 = {6,9}, holding {3,6,9} between them: two cells, three candidates, also almost locked. The shared digits are 3 and 6. The only 3 in A (r1c1) sees the only 3 in B (r1c4) along row 1, so 3 is the restricted common X and 6 becomes Z. Whichever set gives up its 3 locks and must supply the 6 — so a cell that sees both 6s, r2c1 (column 1 to r1c1, row 2 to r2c4), cannot be 6. From two small, harmless-looking sets, a clean elimination appears.
Set A (blue) = {3,6}; set B (green) = {3,9} and {6,9}. The 3s see each other along row 1 (dashed) — 3 is the restricted common. So one set must hold the 6, and r2c1, seeing both 6s, loses its 6.
Step by step
- Find two almost locked sets — each with one more candidate than cells.
- Identify two shared digits, X and Z.
- Confirm X is restricted — every X in one set sees every X in the other.
- Eliminate Z from cells outside both sets that see all copies of Z in each.
Common mistakes
- X not truly restricted. If a single X in one set cannot see an X in the other, the lock fails and no elimination follows.
- Removing Z inside a set. The eliminations are on outside cells only — cells within A or B are part of the argument, not its target.
- Miscounting the set. An almost locked set has exactly one spare candidate; two spares is not the same pattern.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.