What is an ALS?
An Almost Locked Set (ALS) is a small group of empty cells confined to one house (row, column or box) that together contain exactly one more candidate than the number of cells. For example, two cells containing three distinct candidates between them form an ALS(2). ALSes are useful because they behave like nearly-complete mini-groups: if one candidate in the set is false, the remaining cells must resolve the other candidates in restricted ways.
What does ALS-XZ mean?
ALS-XZ is the name given to an elimination pattern that uses two separate ALSes that share exactly two candidates in common — call them X and Z. Those common candidates are called the restricted common candidates. The key idea is logical pressure: if X were eliminated from both ALSes, each ALS would still need to place the remaining candidates, and that can force a contradiction. Therefore a cell that sees (conflicts with) the X-side of one ALS and the Z-side of the other can sometimes be ruled out.
Why this is useful
ALS-XZ allows eliminations that are not visible with simple singles or pairs. Many puzzles leave these forced mini-sets lying around; when two ALSes line up with exactly two common candidates, a candidate outside those sets that sees one common digit in the first ALS and the other common digit in the second ALS can often be proved impossible.
When to look for ALS-XZ
- Mid- to hard-level puzzles where direct eliminations slow down.
- When you spot several small ALSes (size 2 or 3) inside different houses and they share two digits.
- When a candidate appears awkwardly in a cell that sees parts of two ALSes — it’s a prime target to test.
Step-by-step detection (practical method)
1) Find two ALSes confined to different houses (they can touch but must be treated as separate ALSes). Prefer small ALSes (2 or 3 cells).
2) Check their union of candidates. If they share exactly two digits in common (X and Z) and no other digits, you have an ALS pair to test.
3) Identify any candidate outside the ALSes (call it C) that sees at least one instance of X in the first ALS and at least one instance of Z in the second ALS. "Sees" means the cell is in the same row, column or box as those candidate occurrences.
4) If C sees those specific occurrences, C can often be eliminated. The intuitive reason: C being true would block X in one ALS and Z in the other; that forces bad allocations inside both ALSes simultaneously, so C cannot be true.
5) As always, check carefully that the two ALSes really have only X and Z in common and that the seeing relationships cover the necessary occurrences. If everything lines up, remove C and continue solving.
A clear candidate-grid example
Look at the candidate grid in the visual below. Two ALSes are highlighted:
- ALS A (row 2: cells (2,2) and (2,3)) has candidates {1,2,3} across those two cells, so it's an ALS(2).
- ALS B (column 6: cells (4,6) and (7,6)) has candidates {1,3,4} across its two cells, also an ALS(2).
The restricted common candidates are 1 and 3 (these are the X and Z of this example). The cell at (4,3) contains candidate 3 and sees both an occurrence of 1 in ALS A (via column 3 to cell (2,3)) and an occurrence of 3 in ALS B (via row 4 to cell (4,6)). According to ALS-XZ logic, the 3 in (4,3) can be eliminated because selecting it would block the necessary placements of 1 and 3 inside the two ALSes simultaneously.
How to apply this at the table or on paper
- Pencil-mark your ALSes as you find them. It helps to circle the involved cells and note the union candidates.
- When two ALSes share exactly two digits, scan cells that see one of those digits in the first ALS and the other digit in the second ALS. These cells are the elimination targets.
- Verify the seeing relationships: the target must see the specific cells that carry the common digits in the ALSes (not just the house). If the visual alignment is right, make the elimination and re-evaluate.
Common pitfalls
- Make sure the two groups are true ALSes: for k cells they must contain exactly k+1 distinct candidates, all contained in the same house.
- Don’t confuse an ALS pair that shares more than two common digits; ALS-XZ needs exactly two restricted common digits.
- Confirm that the target cell actually “sees” the instances you need. Sometimes a digit appears multiple times in an ALS house; only the instances matter for visibility.
Summary
ALS-XZ is a focused, practical technique for removing candidates when two almost locked sets share exactly two digits. It’s a great tool to add to your intermediate solver’s toolbox: look for small ALSes, check their shared candidates, and scan for a cell that sees the X side of one and the Z side of the other. With a little practice it becomes a quick way to break stalemates and open new solving paths.
Below is a candidate-grid example tied to the explanation — inspect the highlighted ALSes, find the restricted common candidates (1 and 3 here), and confirm why the 3 at (4,3) can be eliminated under the ALS-XZ reasoning.