ALS Chains

What are ALS Chains?

ALS Chains are master-level Sudoku techniques that represent one of the most sophisticated logical structures in puzzle solving. They extend the ALS-XZ concept by linking multiple Almost Locked Sets together through restricted common candidates (RCCs).

Instead of just two ALS interacting (as in ALS-XZ), ALS Chains involve three or more ALS where each consecutive pair is connected by a restricted common. This creates a logical chain that can span across rows, columns, and boxes, enabling eliminations that would be impossible to find with simpler techniques.

Think of it like a chain of dominoes: each ALS forces constraints on the next through their shared RCC. When you trace the chain from beginning to end, candidates that appear at both ends but aren't part of the RCC connections can be eliminated from cells that see both ends.

Why are they called "ALS Chains"?

The name is descriptive: chains (sequences) of Almost Locked Sets connected by logical links. Unlike simple chains of cells, these are chains of multi-cell sets, making them more complex but also more powerful.

They're sometimes called "ALS-Chain-XZ" or "Extended ALS-XZ" in solving literature, emphasizing their relationship to the simpler ALS-XZ technique.

Why They Matter

ALS Chains matter because they:

  • Solve the hardest puzzles — Many diabolical-rated puzzles require ALS Chains or equivalent techniques
  • Unify multiple techniques — Some complex patterns can be viewed as either long forcing chains or ALS Chains
  • Demonstrate logical depth — Show how multiple constrained sets interact across the entire puzzle
  • Provide systematic approach — Instead of random searching, ALS Chains offer structured method
  • Complete the ALS toolkit — Along with ALS-XZ and Sue de Coq, form comprehensive ALS-based solving

Step-by-Step: How to Build an ALS Chain

  1. Find a starting ALS — Identify an Almost Locked Set to begin your chain.
  2. Find a connected ALS — Look for another ALS that shares a restricted common candidate with your first ALS.
  3. Verify the RCC — Confirm that all positions of the common candidate in one ALS see all positions in the other.
  4. Extend the chain — Find additional ALS that connect to your chain through new RCCs.
  5. Identify end candidates — Look for candidates that appear in both the first and last ALS but aren't RCCs in the chain.
  6. Find elimination cells — Locate cells that see all positions of the end candidate in both the first and last ALS.
  7. Eliminate — Remove the end candidate from those cells.

ALS Chain Example

Setup

ALS-1 (Row 2): R2C2={3,7}, R2C5={3,8} → candidates {3,7,8}

ALS-2 (Box 5): R4C4={3,5}, R5C6={5,8} → candidates {3,5,8}

ALS-3 (Column 9): R3C9={5,7}, R7C9={7,9} → candidates {5,7,9}

Chain Connections

  • ALS-1 to ALS-2: RCC = 3 (all 3s in ALS-1 see all 3s in ALS-2)
  • ALS-2 to ALS-3: RCC = 5 (all 5s in ALS-2 see all 5s in ALS-3)

Analysis

Candidate 7 appears in both ALS-1 (R2C2) and ALS-3 (R3C9, R7C9), but 7 is not an RCC in the chain connections. Candidate 8 also appears in ALS-1 and ALS-2 but is not an RCC.

Elimination

Any cell that sees R2C2 (the 7 in ALS-1) AND sees both R3C9 and R7C9 (the 7s in ALS-3) cannot contain 7. The chain forces that 7 must appear in either ALS-1 or ALS-3, eliminating it from cells that see both ends.

Visual Example

Imagine a simplified 3-ALS chain:

  • ALS-A: {2,5,6} in Row 1
  • → RCC: 5 →
  • ALS-B: {4,5,9} in Box 4
  • → RCC: 4 →
  • ALS-C: {2,4,8} in Column 8

Observation: Candidate 2 appears in both ALS-A and ALS-C, but not as an RCC in the chain.

Elimination: Any cell seeing all 2-positions in ALS-A and all 2-positions in ALS-C cannot be 2.

Strategies for Finding ALS Chains

  1. Start with ALS-XZ — Once you find an ALS-XZ, look for ways to extend it into a chain.
  2. Use software assistance — ALS Chain detection is extremely complex; consider solver tools.
  3. Map ALS systematically — Create a list of all ALS in the puzzle and their candidates.
  4. Look for RCC connections — Identify which ALS pairs have restricted common candidates.
  5. Build incrementally — Don't try to see the whole chain at once; add one link at a time.
  6. Practice with examples — Study solved examples before attempting to find chains yourself.

Common Pitfalls

  • Incorrect RCC verification — Each link must have proper RCC restriction. One error breaks the entire chain.
  • Using RCC as elimination candidate — Only candidates that appear at the ends but aren't RCCs can be eliminated.
  • Losing track of complex chains — Chains with 4+ ALS become very difficult to track without notation.
  • Overlapping ALS — ALS in the chain shouldn't share cells (they can be in the same unit but must use different cells).
  • Missing simpler techniques — Always check if shorter chains or simpler techniques work first.
  • Incomplete elimination check — Must verify cells see ALL positions of candidate in both end ALS.

Practice: Identify the Chain

Scenario: You've identified:

  • ALS-1: {1,6,8} in Box 2
  • ALS-2: {1,4,7} in Row 5 (RCC with ALS-1: candidate 1)
  • ALS-3: {4,6,9} in Column 3 (RCC with ALS-2: candidate 4)

Question: What candidate can potentially be eliminated, and from where?

Answer: Candidate 6 appears in both ALS-1 and ALS-3, but 6 is not an RCC in the chain (the RCCs are 1 and 4). Therefore, cells that see all 6-positions in ALS-1 AND all 6-positions in ALS-3 can have 6 eliminated. The chain forces 6 to be in either ALS-1 or ALS-3.

Why ALS Chains Matter

ALS Chains represent the cutting edge of human-solvable Sudoku techniques. They demonstrate that:

  • Complex logical structures can span the entire puzzle grid
  • Multiple constrained sets interact in predictable, exploitable ways
  • Even the hardest puzzles yield to systematic logical analysis
  • Understanding abstract mathematical structures enhances solving capability

While few solvers will apply ALS Chains manually on a regular basis, understanding them provides deep insight into Sudoku's logical structure and completes the toolkit for conquering any puzzle logically.

Quick Recap

Technique How it Works Difficulty
ALS Chains Multiple ALS linked by RCCs, eliminating end candidates from seeing cells Master
ALS-XZ Two ALS connected by RCC enabling Z eliminations Master
Forcing Chains Open chains where paths converge on same conclusion Master

Final Thought

ALS Chains are Sudoku's equivalent of advanced mathematics—beautiful, powerful, and challenging. They prove that even when a puzzle seems impossible, systematic logical structures can unlock the solution. Master them, and you'll have truly conquered Sudoku's logical depths.

Frequently Asked Questions

What are ALS Chains in Sudoku?

ALS Chains are master-level Sudoku techniques that extend ALS-XZ by linking multiple Almost Locked Sets together through restricted common candidates. Instead of just two ALS, you create chains of three or more ALS, each connected to the next, enabling complex eliminations across the puzzle.

How do ALS Chains differ from ALS-XZ?

ALS-XZ uses exactly two ALS connected by one restricted common candidate (X) to eliminate another candidate (Z). ALS Chains extend this concept to three or more ALS, where each consecutive pair shares a restricted common, creating a chain of logical implications that can span large portions of the puzzle.

What is the elimination rule for ALS Chains?

In an ALS Chain, if a candidate appears at both ends of the chain (in the first and last ALS) but is not part of any RCC connections in the chain, you can eliminate that candidate from cells that see all positions of that candidate in both the first and last ALS.

Are ALS Chains practical for manual solving?

ALS Chains are extremely challenging for manual solving due to their complexity—finding multiple ALS, verifying RCC connections, and tracking elimination patterns is demanding. Most solvers use computer assistance to identify ALS Chains, then study the logic to understand the pattern. They're more theoretical than practical for human solving.

How are ALS Chains related to Forcing Chains?

Both involve chaining logical implications across the puzzle. ALS Chains specifically use Almost Locked Sets as the links, while Forcing Chains are more general. ALS Chains can be viewed as a structured type of forcing chain where the nodes are ALS and the connections are restricted commons.

Practice ALS Chains

Explore more advanced techniques: Nice Loops or Kraken Fish.

← Back to All Strategies

All Strategies