ALS-XZ

Recommended first: Candidates · Pairs and Subsets

ALS-XZ sounds abstract, but its central idea is simple:

Two candidate sets that are one step away from being locked are joined by a restricted common candidate X. This guarantees that another common candidate Z is true in at least one set.

To understand that statement, we will build it in order: ALS, RCC, X, Z, and the elimination.

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1. What is an ALS?

ALS means Almost Locked Set.

A group of cells is an ALS when:

1. all cells lie in one row, column, or box and therefore see one another; and
2. N cells contain exactly N+1 distinct candidate digits altogether.

For example, two cells with candidates 1,2 and 2,3 contain three distinct digits: 1,2,3. Two cells and three digits make an ALS.

A locked set has N cells and N digits. Those N digits must fill the N cells, so every digit must occur once. An ALS has just one extra digit. If any one digit is excluded from the whole ALS, N digits remain for N cells and the ALS becomes locked.

Remember an ALS as: N mutually visible cells contain N+1 candidates; remove any one digit from the group and it becomes locked.

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2. What is an RCC?

Take two non-overlapping ALSs, called ALS A and ALS B.

A digit found in both sets is a common candidate, but a common candidate is not automatically an RCC.

RCC means Restricted Common Candidate. A common candidate X is restricted when:

Every position for X in A sees every position for X in B.

Most often, all those X positions share one row, column, or box. That house can contain X only once, so X cannot be true in both ALSs.

This gives the exact RCC conclusion:

• X can be true in at most one ALS;
• at least one ALS therefore does not use X;
• X may also be absent from both ALSs, so a single RCC does not prove that X itself must appear.

The ALS without X loses one candidate and becomes locked. Every remaining digit in that ALS must appear.

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3. How is ALS-XZ formed?

A single-RCC ALS-XZ contains:

1. two non-overlapping almost locked sets A and B;
2. at least two common candidates, called X and Z;
3. X as an RCC, so it cannot be true in both sets;
4. Z as another common candidate; and
5. an outside Z that sees every possible Z position in both A and B.

In the hint diagram:

• blue cells are ALS A;
• yellow cells are ALS B;
• blue candidates are RCC X;
• green candidates are the guaranteed Z positions;
• red candidates are eliminations.

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4. Why can Z be eliminated?

Consider every possible location of X.

X is true in A

B cannot use X. B becomes locked, so every remaining digit in B appears, including Z.

X is true in B

A cannot use X. A becomes locked, so Z appears in A.

Neither ALS uses X

Both sets become locked, so Z appears in both. This only strengthens the conclusion.

In every case:

Z appears at least once in ALS A or ALS B.

An outside Z that sees every Z position in both sets conflicts with Z whichever set supplies it, so that outside candidate can be eliminated.

This is why seeing only some Z positions is insufficient: before solving the pattern, we do not know which ALS will become locked.

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5. Doubly linked ALS-XZ

Sometimes A and B share two RCCs, X and Y. Each RCC must be absent from at least one set.

The same ALS cannot exclude both X and Y. It would lose two digits from an N+1-digit ALS and have only N−1 digits left for N cells, which cannot fill the set.

Therefore the exclusions must split:

• A excludes one RCC;
• B excludes the other RCC;
• both ALSs lose exactly one digit and become locked.

The diagram shows one possible split: A excludes X and B excludes Y, forcing Y into A and X into B. The opposite split works exactly the same way.

This proves stronger conclusions:

1. X and Y each appear exactly once across A and B, one in each set;
2. every non-RCC digit in A is locked into A;
3. every non-RCC digit in B is locked into B.

Blue RCC marks and green non-RCC marks show these guaranteed positions. A red candidate that sees every marked position of its own digit can be eliminated.

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6. Common mistakes

Every common candidate is an RCC

No. Every X in A must see every X in B.

Seeing one Z in each set is enough

No. The elimination must see every possible Z position in both sets.

A single RCC proves X must appear

No. X may be absent from both ALSs. A single-RCC ALS-XZ eliminates the other guaranteed common candidate Z, not X.

The two ALSs may share cells

Not in the ALS-XZ described here. A and B must not overlap.

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7. Reading an ALS-XZ hint

1. Check the blue ALS A and yellow ALS B: each should have N cells and N+1 digits.
2. Check blue X: every X in A must see every X in B.
3. Find common candidate Z in both sets.
4. Confirm that the green marks include every possible Z position in A and B.
5. Confirm that each red candidate sees every green position of the same digit.

When all five checks hold, the elimination follows from the two locked-set possibilities rather than from a guess.