X-Cycles

Level 10 · Brown Belt

Strategy:

Single-Digit Chains

Requires:

Simple Coloring

Learn after:

AICForcing ChainsGrouped X-Cycles

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

Look for X-Cycles.

1

X-Cycles

Build an X-Cycle for digit 2 by following alternating strong and weak links.

2

X-Cycles

A weak link exists when a digit appears in 2+ cells in a row/column/box (they can't both be true).

3

X-Cycles

This cycle has 5 cells with TWO WEAK LINKS meeting at r4c4 (Type 3 discontinuity).

4

X-Cycles

When two weak links meet, the junction cell MUST contain the digit. The highlighted placement is forced at r4c4.

5

X-Cycles

Place 2 at r4c4.

6

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.