X-Cycle

Part 1: Fundamentals and Continuous Loops

By Minimal Sudoku Team•Last updated: February 17, 2026

X-Cycle extends Simple Coloring by combining strong links and weak links on one digit. This page focuses on the core foundation: continuous loops and how they produce eliminations.

Quick Summary

What: A single-digit chain with alternating strong and weak links
Core Pattern: Continuous loops (no flaw) with even length
Main Result: Eliminate candidates that see two opposite same-digit chain endpoints
Progression: Learn this first, then move to discontinuous loops

The Concept

X-Cycle is a chain technique where each candidate is treated as either ON or OFF, and each link carries a logical consequence to the next candidate.

In this Part 1 article, we focus on loops that alternate perfectly all the way around (continuous loops). These are the easiest X-Cycles to trust and apply.

🔄 X-Cycle in One Sentence

Build a one-digit chain that alternates strong and weak links; then eliminate off-chain candidates that see two opposite same-digit endpoints where the chain guarantees at least one endpoint is true.

Core Principle

🎯 Unified Chain Elimination Rule

If two candidates of the same digit are opposite in the chain and the chain guarantees at least one of them is true, any other candidate of that digit that sees both can be removed.

Simple Way to Think About It

You are not eliminating because a line is weak or strong by itself. You are eliminating because the chain gives an either/or pair that always covers the digit.

Strong vs Weak Links

🔗 Strong Link

Exactly 2 cells have the candidate in a unit.

If one is FALSE → other is TRUE

If one is TRUE → other is FALSE

〰️ Weak Link

3+ cells have the candidate in a unit.

If one is TRUE → other is FALSE

If one is FALSE → other is UNKNOWN

Precise Inference View

Strong links support both directions of inference (!A -> B and !B -> A). Weak links support only positive-to-negative inference (A -> !B and B -> !A). This is why strict alternation matters.

Continuous Loops

A continuous X-Cycle has no break in alternation. It must have an even number of nodes, and you can start at any node and traverse either direction.

Continuous loops usually do not place a value directly on a loop node.
They eliminate candidates outside the loop that see both members of an opposite either/or pair on the chain.
This is the same logic family as Simple Coloring traps, but with strong and weak links mixed.

Continuous loop notation example (digit 9):

+9[B3]-9[B8]+9[H8]-9[H3]+9[B3]

This is a continuous X-Cycle (X-Wing structure):
- Strong and weak links alternate all the way around
- The loop closes cleanly with no discontinuity
- Use it to remove off-chain 9s that see both relevant loop nodes

✅ Continuous Loop Outcome

In a continuous loop, opposite endpoints create either/or truth pairs. Any off-chain candidate that sees both endpoints of such a pair is impossible and can be removed.

Worked Continuous Example

Continuous X-Cycle on Digit 3 (8-node loop)

Continuous X-Cycle with 8 loop nodes on candidate 3

This position shows a continuous X-Cycle on candidate 3 with 8 loop nodes (an even-length loop).

Because the chain alternates cleanly with no discontinuity, it acts as an elimination engine: off-chain 3-candidates can be removed whenever they see both members of an opposite either/or pair on the chain.

Result

This loop removes multiple candidate 3 pencil marks outside the chain, which is exactly the expected outcome of a continuous X-Cycle.

Classic Patterns Inside X-Cycle

Some named patterns are just short continuous X-Cycles:

X-Wing: continuous X-Cycle with 4 nodes.
2-2-2 Swordfish: continuous X-Cycle with 6 nodes.
Thinking in X-Cycles helps you see these as one consistent chain framework.

How to Find X-Cycles

Choose one digit

Pick a candidate value and inspect all of its occurrences across the grid.

Mark conjugate pairs first

Start with strong links (exactly two candidates in a unit). They are your most reliable anchors.

Connect with weak links

Add weak links only between candidates that see each other for the same digit.

Check alternation and parity

Verify strong/weak alternation and confirm the loop length is even for continuous patterns.

Harvest off-chain eliminations

Remove candidates outside the loop that see both members of an opposite same-digit pair guaranteed to contain the truth.

Common Mistakes

Forgetting Visibility

A weak link requires both candidates to see each other in the same row, column, or box. Shared digit alone is not enough.

Trying to Place from Every Loop

Continuous loops are primarily elimination tools. Direct placements come from discontinuous loops, covered in Part 2.

Mark S/W While Tracing

Label each edge as S or W while building a chain. This catches invalid transitions early.

Next Step

After continuous loops, the next skill is discontinuous X-Cycles. That is where you get direct placements (strong-strong) and discontinuity-node eliminations (weak-weak).

Chain Techniques

🎨Simple Coloring 🧩X-Cycle Discontinuous Loops 🔗XY-Chain 🐍3D Medusa

Expert

X-Cycle Discontinuous Loops

Learn the two discontinuity outcomes: strong-strong placement and weak-weak elimination.