ALS-XZ

What is ALS-XZ?

ALS-XZ is a master-level Sudoku technique that harnesses the power of Almost Locked Sets to create eliminations. It involves two ALS connected by shared candidates in a specific way: one candidate (X) acts as a Restricted Common Candidate (RCC), while another candidate (Z) can be eliminated from cells that see both ALS.

The beauty of ALS-XZ is in its logical forcing: when two ALS share a restricted common, one of them must "lock" and contain certain candidates. This forces eliminations in a way that simpler techniques cannot achieve.

Think of it like a logical seesaw: if candidate X goes into ALS-A, then ALS-B must contain Z. If X goes into ALS-B instead, then ALS-A must contain Z. Either way, one of the ALS definitely contains Z, so you can eliminate Z from cells that see all the Z-positions in that ALS.

Why is it called "ALS-XZ"?

The name comes from the three key elements:

  • ALS — Two Almost Locked Sets are required
  • X — The Restricted Common Candidate that connects the two ALS
  • Z — The candidate that can be eliminated (often called the "elimination candidate")

Sometimes you'll see it written as "ALS-XZ Rule" or just "XZ-Rule" in solving literature.

Why It Matters

ALS-XZ matters because it:

  • Unlocks difficult puzzles — Many expert-level puzzles require ALS-XZ or similar techniques
  • Generalizes simpler patterns — Some XY-Wing and X-Wing patterns are special cases of ALS-XZ
  • Demonstrates ALS power — Shows how ALS interactions create eliminations
  • Enables further techniques — Understanding ALS-XZ is essential for ALS Chains and other advanced methods
  • Provides systematic approach — Offers a structured method for finding complex eliminations

Step-by-Step: How to Find ALS-XZ

  1. Find two Almost Locked Sets — Identify two ALS in different units (they can be in different rows, columns, or boxes).
  2. Identify common candidates — List candidates that appear in both ALS.
  3. Check for Restricted Common (X) — Find a candidate where all cells containing it in ALS-A see all cells containing it in ALS-B.
  4. Identify elimination candidate (Z) — Choose another common candidate (not the RCC) as your elimination target.
  5. Find cells that see all Z-positions — Look for cells that can see every cell containing Z in one of the ALS.
  6. Eliminate Z — Remove Z from those cells.

ALS-XZ Example

Setup

ALS-A (in Row 2): R2C3={3,7} and R2C5={3,8} — 2 cells, 3 candidates {3,7,8}

ALS-B (in Box 1): R1C2={3,5} and R3C2={5,8} — 2 cells, 3 candidates {3,5,8}

Analysis

  • Common candidates: {3,8} appear in both ALS
  • RCC check (candidate 3): In ALS-A, 3 is in R2C3 and R2C5. In ALS-B, 3 is in R1C2. All positions see each other (R2C3 and R2C5 both see R1C2 in Box 1). So 3 is a valid RCC.
  • Elimination candidate: Z = 8 (the other common candidate)

Logic

If 3 goes into ALS-A (either R2C3 or R2C5), then ALS-B cannot have 3 in R1C2, forcing ALS-B to lock with {5,8}. This means ALS-B must contain 8.

If 3 goes into ALS-B (R1C2), then ALS-A cannot have 3, forcing ALS-A to lock with {7,8}. This means ALS-A must contain 8.

Either way, one of the ALS must contain 8. Therefore, any cell that sees all 8-positions in ALS-A (R2C5) cannot be 8. Eliminate 8 from cells that see R2C5.

Visual Example

Consider this pattern:

  • ALS-A: Three cells in Column 4 containing {2,6,9} total
  • ALS-B: Two cells in Row 5 containing {2,6,7} total
  • Common candidates: {2,6}
  • RCC: Candidate 2 (all 2s in ALS-A see all 2s in ALS-B)
  • Z-candidate: 6

Elimination: Since one ALS must contain 6, any cell that sees all 6-positions in ALS-B cannot be 6.

Strategies for Finding ALS-XZ

  1. Start by finding ALS — Systematically identify ALS in rows, columns, and boxes.
  2. Look for ALS with common candidates — Focus on ALS that share 2-3 candidates.
  3. Verify RCC carefully — Check that all positions of the RCC in one ALS see all positions in the other.
  4. Use candidate highlighting — Color-coding makes seeing relationships easier.
  5. Check common candidates systematically — Test each common candidate to see if it qualifies as an RCC.
  6. Practice with software — Use Sudoku solvers that highlight ALS-XZ to learn pattern recognition.

Common Pitfalls

  • Incorrect RCC verification — The most common error is assuming a candidate is restricted when positions don't all see each other.
  • Confusing X and Z — The RCC (X) is the connector; Z is the elimination candidate. Don't mix them up.
  • Missing elimination cells — Remember to eliminate from cells that see ALL Z-positions in the target ALS, not just some.
  • Overlapping ALS — ALS must be in different units or have non-overlapping cells for valid ALS-XZ.
  • Forgetting to check both Z-directions — Sometimes you can eliminate Z from cells seeing ALS-A, sometimes from cells seeing ALS-B, sometimes both.

Practice: Find the ALS-XZ

Scenario:

  • ALS-A (Box 3): R2C7={4,9}, R2C9={4,5} → candidates {4,5,9}
  • ALS-B (Row 8): R8C2={4,7}, R8C4={5,7} → candidates {4,5,7}
  • Common candidates: {4,5}

Question: Which candidate is the RCC (X), and what can you eliminate?

Answer: First check if 4 is restricted: In ALS-A, 4 is in R2C7 and R2C9. In ALS-B, 4 is in R8C2. Do R2C7 and R2C9 both see R8C2? R2C7 sees R8C2 (same column? No. Same box? No. Same row? No). They don't all see each other, so 4 is not an RCC. Now check 5: In ALS-A, 5 is in R2C9. In ALS-B, 5 is in R8C4. If these see each other and all positions are verified as restricted, 5 is the RCC (X). Then 4 becomes Z, and you can eliminate 4 from cells that see all 4-positions in one of the ALS.

Why ALS-XZ Matters

ALS-XZ represents a significant step up in Sudoku solving sophistication. It demonstrates how:

  • Simple building blocks (ALS) combine to create powerful patterns
  • Restricted relationships enable distant eliminations
  • Forcing logic can prove candidates must exist in certain regions
  • Multiple seemingly unrelated parts of the puzzle can interact logically

Mastering ALS-XZ opens the door to even more advanced ALS-based techniques and provides a powerful tool for conquering expert-level puzzles.

Quick Recap

Technique How it Works Difficulty
ALS-XZ Two ALS connected by RCC (X) force eliminations of candidate Z Master
ALS N cells with N+1 candidates (building block for advanced techniques) Master
XY-Wing Three bi-value cells creating elimination pattern Advanced

Final Thought

ALS-XZ is where Sudoku solving becomes truly mathematical—abstract sets interacting through restricted relationships to force logical conclusions. It's challenging but deeply rewarding, and once mastered, it reveals the elegant mathematical structure underlying even the hardest puzzles.

Frequently Asked Questions

What is ALS-XZ in Sudoku?

ALS-XZ is a master-level Sudoku technique that uses two Almost Locked Sets (ALS) connected by restricted common candidates. When two ALS share candidates X and Z in specific ways, you can eliminate candidate Z from cells that see all Z-positions in one of the ALS. It's a powerful pattern for difficult puzzles.

What is a Restricted Common Candidate (RCC)?

A Restricted Common Candidate (RCC) is a candidate that appears in both ALS but is restricted to seeing positions. Specifically, every cell containing the RCC in ALS-A must see every cell containing the RCC in ALS-B. This restriction is what enables ALS-XZ eliminations.

How does ALS-XZ create eliminations?

ALS-XZ works by forcing one of the ALS to 'give up' a candidate. If the RCC (X) goes in ALS-A, then ALS-B becomes locked and must contain Z. If X goes in ALS-B instead, ALS-A becomes locked and must contain Z. Either way, one ALS must contain Z, eliminating Z from cells that see all Z-positions in that ALS.

What's the difference between ALS-XZ and Sue de Coq?

Both use ALS interactions, but Sue de Coq is a specific configuration at box-line intersections. ALS-XZ is more general—it works with any two ALS in different units connected by restricted commons. Sue de Coq can be viewed as a special case of ALS logic with additional geometric constraints.

Is ALS-XZ practical for manual solving?

ALS-XZ is challenging for manual solving because it requires identifying two ALS, verifying RCC restrictions, and tracking elimination patterns. However, with practice and systematic searching, it becomes manageable. Many solvers use software to find ALS-XZ patterns, then verify the logic manually.

Practice ALS-XZ

Ready for more ALS techniques? Try ALS Chains or Sue de Coq.

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