X-Cycles

Level 11 · Red Belt

Strategy:

Single-Digit Chains

Requires:

Simple Coloring

Learn after:

Forcing Chains

Overview

X-Cycles are chains of strong and weak links for a single digit that form loops. Depending on link arrangement, eliminations or placements can be determined.

How It Works

  1. Build a chain using strong links (exactly 2 in unit) and weak links (more than 2 in unit).
  2. Even loop (alternating strong-weak): Eliminate the candidate from cells seeing two chain cells.
  3. Odd loop with two strong links adjacent: The candidate between them must be true.
  4. Odd loop with two weak links adjacent: The candidate between them must be false.

Example

Starting position

Look for X-Cycles.

1

Build an X-Cycle on digit 6

Build an X-Cycle for digit 6 by following alternating strong and weak links.

2

Type 1 nice loop of 4 cells

This is a 'Nice Loop' of 4 cells with perfectly alternating links (Type 1).

3

One end of each strong link is true

In a nice loop, exactly one endpoint of each strong link is true.

4

Cells seeing both ends lose 6

Cells seeing both endpoints of a strong link cannot contain 6. These highlighted candidates are affected: .

5

Remove 6 from r1c9

Remove 6 from r1c9.

6

Result

Result after applying the technique.

Tips & Patterns

  • X-Cycles generalize Skyscrapers, X-Wings, and other single-digit patterns.
  • Strong links force truth; weak links force falsehood.
  • Even loops confirm the chain; odd loops find contradictions.