ALS-XZ: Almost Locked Sets, Explained

ALS-XZ is the most abstract technique here, and it's fair to arrive at this one only after everything else — XY-Wing, W-Wing, and XYZ-Wing among them — is genuinely familiar. It rewards that familiarity: you've actually been using a tiny version of it already, just without the name.

First, what's an "almost locked set"?

A locked set is N cells in one row, column, or box that between them hold exactly N candidates — a Naked Pair is a locked set of two, a Naked Triple of three. An almost locked set (ALS) is the same idea, one candidate short of locked: N cells that between them hold N+1 candidates — one more digit than cells, so it's one placement away from becoming a locked set.

What is ALS-XZ?

Take two almost locked sets that share a digit X, where every copy of X in the first set sees every copy of X in the second — a "restricted common." That means the two sets can't both avoid placing X somewhere in the overlap: they can't both be X, so whichever one isn't, uses up all of its own cells and candidates — including any other digit Z the two sets have in common.

So one of the two sets always places a Z somewhere inside it. Any cell that sees every Z-candidate cell in both sets can't be Z.

Why "restricted common" is the hard part

The requirement is that every cell holding X in set A sees every cell holding X in set B — not just some of them. Get this wrong and the whole deduction falls apart, because then it's possible for both sets to place X without contradiction, and the "one of them must use Z" argument no longer holds. When X appears in only one cell on each side, the condition is automatic — those are the easiest instances to find.

A worked example

Here's the exact board the app uses for its own ALS-XZ lesson:

Singles are the only easier technique this puzzle needs — they clear a lot, but not quite everything:

The two almost locked sets that finish it sit in row 1 and row 8. ALS-A is two cells in row 1 holding {1, 4, 7} between them; ALS-B is three cells in row 8 holding {1, 2, 4, 7}:

Blue & violet: the cells the logic runs on · Rose: where the elimination lands

Both sets share the digit 7, and the only 7 in each set — R1C9 and R8C9 — sit in the same column, so they see each other: a valid restricted common. Both sets also share the digit 4, appearing only at R1C2 in ALS-A and only at R8C3 in ALS-B. Since the two sets can't both be 7, one of them must place every one of its own candidates — including a 4. Any cell that sees both R1C2 and R8C3 can't be a 4 — here, R3C3 (shares a box with R1C2, a column with R8C3) and R9C2 (shares a column with R1C2, a box with R8C3).

How to spot one

1. Look for small groups of cells in a single unit — even one cell with two candidates counts — that hold exactly one more digit than they have cells.
2. Find two such groups sharing at least two digits.
3. Check whether the copies of one shared digit all see each other — that's your restricted common (X).
4. For the other shared digit (Z), eliminate it from any cell that sees every Z-cell in both groups.