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 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.

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).

Type 1 (Continuous Nice Loop): Even length, perfect alternation. The loop acts like a closed coloring chain. Eliminate the digit from any uncolored candidate that sees both colors, and eliminate candidates that conflict with the loop's confirmed link structure.

Type 2 (Two Strong Links Meet): A discontinuity where two strong links meet at the same candidate forces that candidate true. Place the digit in the junction cell.

Type 3 (Two Weak Links Meet): A discontinuity where two weak links meet at the same candidate makes that candidate false. Eliminate the digit from the junction cell.

Level 10 (Master).

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.

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.

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.

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 that at least one endpoint is true. Closed loops produce eliminations or placements depending on whether they are continuous or where the discontinuity occurs.