Single-Digit Chains

What Are Single-Digit Chains?

A logical path following one digit through candidate positions, alternating strong and weak links. They bridge the gap between fixed-shape single-digit patterns and general alternating inference chains.

Strong Links and Weak Links

Strong link: two cells are the only positions for a digit in a house. If one is false, the other must be true. Weak link: two cells share a house and both contain the digit. If one is true, the other must be false. The power comes from alternating these link types.

X-Chain: The Open Chain

A sequence alternating strong and weak links, starting and ending on strong links. At least one endpoint must be true. Eliminate the digit from cells seeing both endpoints. A length-3 X-Chain is equivalent to Skyscraper, Two-String Kite, or Crane. Level 9 (Master).

X-Cycles: The Closed Chain

Type 1 (Continuous Nice Loop): Even length, perfect alternation. Eliminate from cells seeing both ends of any strong link in the loop. Type 2 (Two Strong Links Meet): Odd length. Eliminate the digit from the junction node. Type 3 (Two Weak Links Meet): Odd length. Place the digit in the junction cell. Level 10 (Master).

Grouped X-Cycles

Groups of cells at block-line intersections act as single nodes. The three cycle types apply identically. Level 11 (Extreme). Catches patterns invisible to standard X-Cycles.

How to Find Single-Digit Chains

1. Map all strong links for a digit. 2. Build chains by alternating strong and weak links. 3. Check for useful results (open chain endpoints or closed loop types). 4. Consider group nodes if standard chains fail.

Relationship to Other Techniques

Skyscraper, Two-String Kite, Crane are all length-3 X-Chains. Empty Rectangle uses an implicit strong link through a block. Single-digit chains are the gateway to AICs. Fish patterns can be expressed as overlapping X-Chains.

Summary

X-Chain (Level 9), X-Cycles (Level 10), Grouped X-Cycles (Level 11). All share the same principle: alternating strong and weak links on a single digit. Open chains prove one endpoint true. Closed loops produce eliminations or placements depending on type.