X-Cycle Discontinuous Loops

Part 2: Placements and Eliminations at the Discontinuity

By Minimal Sudoku Team•Last updated: February 16, 2026

This is Part 2 of X-Cycle. If you have not covered fundamentals yet, start with X-Cycle Part 1. Here we focus on loops with exactly one flaw in alternation, called a discontinuity.

Quick Summary

Structure: One odd-length loop with exactly one discontinuity
Strong-Strong: Candidate at the discontinuity is forced TRUE (placement)
Weak-Weak: Candidate at the discontinuity is forced FALSE (elimination)
Use Case: Great for stalled expert grids where continuous loops find nothing

The Concept

A discontinuity means alternation breaks at one node: either two strong links touch, or two weak links touch.

For clean X-Cycle inference, the loop should have exactly one discontinuity. That single flaw creates a forced conclusion at the discontinuity node.

🧩 Two Outcomes, One Pattern

At the discontinuity node, strong-strong means the candidate must be true. Weak-weak means the candidate must be false.

Parity and Loop Shape

Perfectly alternating (continuous) loops are even-length.
Single-discontinuity loops are odd-length.
If you detect multiple discontinuities, re-check your links before applying conclusions.

Learning Shortcut

Start tracing from the discontinuity node. It makes the contradiction easier to see and keeps chain notation consistent.

Strong-Strong Discontinuity

When two strong links meet at one node, assume that node is false and walk the chain. The chain forces conflicting truths, so the assumption fails.

Placement Rule

Strong-strong at the discontinuity forces the node to be TRUE. Place the digit there.

Weak-Weak Discontinuity

When two weak links meet at one node, assume that node is true and walk the chain. The chain returns a contradiction, so the node cannot be true.

Elimination Rule

Weak-weak at the discontinuity forces the node to be FALSE. Remove that candidate from the node.

Weak-weak example notation (digit 1):

+1[C3]-1[C7]+1[G7]-1[G2]+1[H3]-1[C3]

If C3 is set to 1, the chain returns to C3 as not-1.
Contradiction -> remove 1 from C3.

Worked Example

X-Cycle (Placement) on Digit 8

Discontinuous X-Cycle placement where two strong links meet at R5C4

This grid shows a discontinuous X-Cycle on digit 8 with a strong-strong discontinuity.

Two strong links meet at R5C4. If R5C4 were not 8, the chain would force an impossible configuration elsewhere in the loop.

Placement

A discontinuous chain on digit 8 with two strong links meeting at R5C4 proves this cell must be 8.

How to Detect Quickly

Build from strong links first

Start as in Part 1: choose one digit and map conjugate pairs.

Track S/W labels explicitly

Mark every edge as strong or weak while extending the chain.

Look for one odd-node loop

A single odd loop with one S-S or W-W junction is your discontinuous target.

Test the junction with contradiction logic

Assume the opposite truth at the junction and verify whether the chain breaks Sudoku constraints.

Apply only the local conclusion

For discontinuous loops, the conclusion is at the discontinuity node itself: place or eliminate.

Avoid Overreach

Do not mix continuous-loop eliminations and discontinuity-node conclusions in the same move unless each inference is independently valid.

Chain Techniques

🔄X-Cycle Fundamentals (Part 1)🎨Simple Coloring 🔗AIC 🧠XY-Chain

Advanced

X-Cycle Fundamentals

Review continuous loops, core link logic, and off-chain elimination patterns.

Expert

Alternating Inference Chains

Generalize these discontinuity ideas to multi-digit chain logic.