ALS-XZ is the simplest major technique built directly from Almost Locked Sets.
It combines two ALSs using a Restricted Common Candidate X. If both ALSs also contain another shared digit Z, then at least one of the ALSs must contain Z.
Any outside candidate that sees all possible Z occurrences across both ALSs can therefore be eliminated.
That is the singly linked ALS-XZ rule.
Prerequisite: two valid ALSs
Start with two sets:
- ALS A = N cells / N+1 candidates in one house;
- ALS B = M cells / M+1 candidates in one house.
Each ALS may be as small as one bivalue cell.
Do not search for X or Z until both sets independently pass the ALS definition.
X: the Restricted Common Candidate
Candidate X occurs in both ALS A and ALS B.
For X to be an RCC, its occurrences must be restricted so that X cannot be true in both ALSs simultaneously.
The usual reason is that all relevant X occurrences across the two sets share a common house relationship.
Therefore:
- if X is placed in A, X disappears from B;
- B then loses one candidate and becomes a Locked Set;
- if X is placed in B, the reverse happens.
At least one ALS becomes locked through the X relationship.
Z: the shared elimination candidate
Now suppose both ALSs also contain digit Z, and Z is not serving as the single RCC X in the ordinary singly linked case.
Whichever ALS becomes locked must contain its remaining required digits, including a Z.
Therefore:
Z must be true somewhere in ALS A or ALS B.
A candidate Z outside the ALSs that sees every possible Z location in both sets cannot be true.
Worked ALS-XZ
The proof in two cases
There are two possibilities for the RCC X.
X is true in ALS A
X cannot be true in ALS B. Remove X from B conceptually.
B now has M cells / M candidates and becomes locked. Because Z is among those candidates, B contains a true Z somewhere.
X is not used in ALS A / is true in B
Symmetrically, A becomes locked and contains Z somewhere.
So in every valid solution, one of the two ALSs contains Z.
The target seeing all Z possibilities is false in both cases.
Singly linked ALS-XZ
The introductory form uses one RCC X.
The main elimination is on another shared digit Z.
This is the cleanest form to learn because the two-case proof is direct and target visibility is easy to state.
Doubly linked ALS-XZ
Two ALSs may have two RCCs.
This is more powerful. The two restricted digits cannot both live in the same ALS in a way that would leave the other set under-supplied, so both ALSs gain stronger locked-set consequences.
Possible eliminations can include:
- the RCCs from other cells in their shared houses;
- non-RCC candidates that become locked inside one ALS;
- multiple candidates rather than one Z.
Doubly linked ALS-XZ is part of this same URL because it is a direct extension of the rule, not a separate search intent for our freeze.
ALS-XZ vs WXYZ-Wing
Many WXYZ-Wings can be expressed as ALS-XZ.
The Wing is a named, visually recognizable four-cell special case. ALS-XZ allows arbitrary ALS sizes and candidate distributions as long as the ALS/RCC conditions are valid.
If you can see the Wing immediately, use it. If the structure is irregular, ALS-XZ gives the more general proof.
ALS-XZ vs XY-Wing
XY-Wing can also be understood through small ALS structures: bivalue cells are size-1 ALSs.
Again, the named pattern is easier to spot, while ALS provides the general framework.
How to search for ALS-XZ efficiently
- find small ALSs first;
- pair ALSs that share at least one candidate;
- test whether one shared digit is restricted enough to be an RCC;
- look for another common candidate Z;
- map all Z locations;
- search only cells that see all Z locations.
A good human search usually starts from an obvious bivalue/trivalue cluster and looks outward rather than enumerating every ALS pair.
Common mistakes
X is common but not restricted
Then it is not an RCC and the two-case lock proof fails.
Z is visible in only part of one ALS
The target must see every possible Z occurrence relevant to the forced statement.
One “ALS” is actually N cells / N+2 candidates
Then it is not Almost Locked.
The target belongs to the ALS without a valid cannibalistic proof
Standard introductory ALS-XZ targets are outside the sets. Advanced cannibalistic eliminations need explicit validation.
Confusing doubly linked rules with singly linked rules
Two RCCs create additional conclusions; do not apply them when only one RCC exists.
FAQ
What do X and Z mean in ALS-XZ?
X is the Restricted Common Candidate connecting the two ALSs. Z is another shared candidate that is guaranteed to occur in at least one ALS and therefore can be eliminated from common peers.
How many ALSs does ALS-XZ use?
Two.
Can a bivalue cell be one of the ALSs?
Yes. A bivalue cell is a size-1 ALS.
What is doubly linked ALS-XZ?
It is an ALS-XZ relationship where the two ALSs have two RCCs, producing stronger locked-set eliminations.
What to learn next
The natural later extensions are ALS-XY-Wing and ALS Chains. They are intentionally left for post-freeze expert expansion; ALS-XZ is the foundational model to master first.