Grouped X-Cycles

Level 11 · Red Belt

Strategy:

Single-Digit Chains

Overview

Grouped X-Cycles extend X-Cycles by allowing grouped nodes - where multiple cells in a box-line intersection act as a single node. This enables stronger chains and more elimination opportunities.

How It Works

  1. Build a chain for a single digit using individual cells and grouped nodes.
  2. A grouped node represents all occurrences of the digit in a box-line intersection.
  3. Strong links connect a node to its conjugate; weak links connect nodes sharing a house.
  4. Apply the same loop rules as X-Cycles: continuous loops eliminate; discontinuous loops force or eliminate.

Example

Starting position

Look for Grouped X-Cycles.

1

Grouped X-Cycle on 2

Found a Grouped X-Cycle on digit 2.

2

Loop of 4 grouped nodes

The cycle has 4 grouped nodes with alternating strong and weak links.

3

Peers of both link ends lose the digit

Any cell seeing both endpoints of a strong link in the loop can have this digit removed.

4

These cells cannot keep 2

These highlighted cells see both ends of a strong grouped link and cannot keep 2: .

5

Remove 2 from r9c2 and r9c3

Remove 2 from r9c2 and r9c3.

6

Result

Result after applying the technique.

Tips & Patterns

  • Grouped nodes represent 2-3 cells acting as one unit.
  • The group is 'on' if any cell in it contains the digit.
  • This is a natural extension of X-Cycles that handles more complex single-digit patterns.