Grouped X-Cycles

Grouped X-Cycles extend X-Cycles by treating multiple aligned cells as a single logical node. When 2-3 cells align within a box (forming pointing pairs or triples), they can act as one unit in the chain, enabling eliminations impossible with standard X-Cycles.

Note: This technique uses chain colouring to display candidates. Square (▢) and Diamond (◇) shapes represent the two colour partitions.

Link Types Review

Before exploring grouped nodes, recall the two link types:

Strong Link (Conjugate Pair)

Two cells in a unit where the digit can only appear in those two positions.

If one is FALSE, the other must be TRUE
Notation: A ═══ B

Weak Link

Two cells in the same unit with the digit as a candidate.

If one is TRUE, the other must be FALSE
Notation: A ─ ─ ─ B

Grouped Nodes

In standard X-Cycles, each node is a single cell. Grouped X-Cycles introduce a powerful extension:

What Is a Grouped Node?

A grouped node consists of 2-3 cells in the same box, aligned in a single row or column, all containing the target digit as a candidate. These cells act as a single logical unit in the chain.

Box containing grouped node:
┌───────┬───────┬───────┐
│  .    │  .    │  .    │
│  .    │  .    │  .    │
│ [2]   │ [2]   │  .    │  ← R3C1 and R3C2 form a grouped node
└───────┴───────┴───────┘

Why Grouping Works

When cells are aligned within a box:

They form a "pointing" pattern to the rest of their row/column
If the digit appears in ANY of them, it's eliminated from the rest of that line
Exactly ONE cell in the group will contain the digit (or none will)
The group acts as a single logical unit in the chain

This means:

A group can form a strong link with a cell outside the box (in the same row/column)
A group can form a weak link with cells in the same box or along the same line
Links to/from the group apply to ALL cells in the group simultaneously

Identifying Grouped Nodes

Look for aligned candidates within a box:

Find a digit with 2-3 candidates in a box
Check if those candidates are in the same row OR column
If aligned, they can be treated as a grouped node

Cycle Types and Rules

Grouped X-Cycles use the same rules as regular X-Cycles:

Rule 1: Continuous Loop (Nice Loop)

A closed chain with alternating strong and weak links throughout.

Elimination: Any candidate that sees two cells connected by a weak link can be eliminated.

Rule 2: Strong Junction

A discontinuous loop where two strong links meet at one node.

Result: The junction node must be TRUE (place the digit).

Rule 3: Weak Junction

A discontinuous loop where two weak links meet at one node.

Result: The junction node must be FALSE (eliminate the digit).

Example: Continuous Loop (Rule 1)

This example demonstrates a Continuous Loop on digit 7. The chain includes TWO grouped nodes, enabling off-chain eliminations that would be impossible with standard X-Cycles.

Pattern Analysis:

Digit: 7
Chain cells: R2C3, R2C7, R3C8, R6C1 (individual) + [R2C1, R3C1] + [R4C8, R6C8] (grouped)
Grouped nodes:
- [R2C1, R3C1] — aligned in column 1, box 1
- [R4C8, R6C8] — aligned in column 8, box 6
Eliminations: R4C1~~7, R5C1~~7, R8C1~~7, R9C8~~7

puzzle: S9B024Z041F1J070953056R095V051J0237045337030504090802371F5Z5V065V0209052F046F6B026R041F37091J2B040903051F0837025F5F030907041F024ZDZ025Z4Z531J3F056Z04776B024Z1V036R09
mode: guided
technique: Grouped X-Cycles
initial:
  layers:
    hints: true
steps:
  - text: >
      Grouped X-Cycles use the same rules as X-Cycles but allow treating aligned cells as a single node. Look for digit 7 — it has aligned candidates in boxes 1 and 6 that can form grouped nodes.
    hint: subtle
    technique: GXC

  - text: >
      Enable the chain display for digit 7. Notice how some cells are grouped together — these aligned candidates can act as single logical units.
    hint: subtle
    technique: GXC
    state:
      graph:
        type: singleDigitChain
        digits: [7]

  - text: >
      Start at R2C3 and assign it ◇ (Diamond). R2C3 has a strong link to the grouped node [R2C1, R3C1]. After a strong link, the colour switches — so the group gets ▢ (Square).
    hint: subtle
    technique: GXC
    state:
      selection:
        cells: [R2C1, R3C1, R2C3]
      graph:
        type: singleDigitChain
        digits: [7]

  - text: >
      The group [R2C1, R3C1] (▢) has a weak link to R6C1. After a weak link, the colour also switches — R6C1 gets ◇. Then R6C1 (◇) has a strong link to [R4C8, R6C8], switching to ▢.
    hint: subtle
    technique: GXC
    state:
      selection:
        cells: [R2C1, R3C1, R4C8, R6C1, R6C8]
      graph:
        type: singleDigitChain
        digits: [7]

  - text: >
      Continue: [R4C8, R6C8] (▢) connects weakly to R3C8 — switch to ◇. Then R3C8 (◇) has a strong link to R2C7 — switch to ▢. Finally R2C7 (▢) connects weakly to R2C3.
    hint: obvious
    technique: GXC
    state:
      selection:
        cells: [R2C1, R3C1, R2C3, R2C7, R3C8, R4C8, R6C1, R6C8]
      graph:
        type: singleDigitChain
        digits: [7]

  - text: >
      After the weak link from R2C7 (▢), we return to R2C3 with ◇ — exactly what we started with! The colours alternate perfectly through all 6 links, confirming a valid continuous loop.
    hint: obvious
    technique: GXC
    state:
      selection:
        cells: [R2C1, R3C1, R2C3, R2C7, R3C8, R4C8, R6C1, R6C8]

  - text: >
      Rule 1 applies: eliminate 7 from cells that see both endpoints of a weak link. R4C1 and R5C1 see both the group [R2C1, R3C1] (in column 1) and R6C1 (also in column 1). Eliminate 7 from these cells.
    hint: detailed
    technique: GXC
    state:
      selection:
        cells: [R4C1, R5C1, R8C1]

  - text: >
      Similarly, R9C8 sees both R3C8 and the group [R4C8, R6C8]. All off-chain candidates caught between weak links are eliminated: R4C1~7, R5C1~7, R8C1~7, R9C8~7.
    hint: detailed
    technique: GXC
    state:
      selection:
        cells: [R4C1, R5C1, R8C1, R9C8]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  coordinateFormat: rncn
  navigation: numbered

The power of grouped nodes: this continuous loop would NOT be possible with standard X-Cycles because individual cells wouldn't form the necessary strong links.

Example: Weak Junction (Rule 3)

This example demonstrates a Weak Junction pattern on digit 2. The grouped node enables a chain that eliminates a candidate at the junction where two weak links meet.

Pattern Analysis:

Digit: 2
Chain cells: R2C3, R2C9, R7C1, R7C9 (individual) + [R2C1, R3C1] (grouped)
Grouped node: [R2C1, R3C1] — aligned in column 1, box 1
Junction: R2C9 (two weak links meet)
Elimination: R2C9~2

puzzle: S9B0108057U020603077U0W060WA60V334S44BW0W09071A0U08011G246Y014A4E05024Y096UDO2CBG4E060U4K496H5G03440107095G045G60040106462E090548D42CBC6M4F3358555D46050602090V07474F
mode: guided
technique: Grouped X-Cycles
initial:
  layers:
    hints: true
steps:
  - text: >
      Rule 3 eliminates candidates at a junction where two weak links meet. Look for digit 2 — it has a grouped node in box 1 and a potential weak junction elsewhere.
    hint: subtle
    technique: GXC

  - text: >
      Enable the chain display for digit 2. Find the grouped node: R2C1 and R3C1 are aligned in column 1 within box 1, both containing candidate 2.
    hint: subtle
    technique: GXC
    state:
      selection:
        cells: [R2C1, R3C1]
      graph:
        type: singleDigitChain
        digits: [2]

  - text: >
      Start at R2C3 and assign it ◇ (Diamond). R2C3 has a strong link to [R2C1, R3C1]. After the strong link, switch colours — the group gets ▢ (Square).
    hint: subtle
    technique: GXC
    state:
      selection:
        cells: [R2C1, R3C1, R2C3]
      graph:
        type: singleDigitChain
        digits: [2]

  - text: >
      The group [R2C1, R3C1] (▢) connects weakly to R7C1. After the weak link, switch colours — R7C1 gets ◇. Then R7C1 (◇) has a strong link to R7C9, switching to ▢.
    hint: obvious
    technique: GXC
    state:
      selection:
        cells: [R2C1, R3C1, R2C3, R7C1, R7C9]
      graph:
        type: singleDigitChain
        digits: [2]

  - text: >
      R7C9 (▢) connects weakly to R2C9. After the weak link, switch colours — R2C9 should get ◇. But wait: R2C9 also has a weak link to R2C3 (◇), which would make it ▢!
    hint: obvious
    technique: GXC
    state:
      selection:
        cells: [R2C1, R3C1, R2C3, R2C9, R7C1, R7C9]
      graph:
        type: singleDigitChain
        digits: [2]

  - text: >
      R2C9 is the junction of two weak links: from R7C9 (▢) it should be ◇, but from R2C3 (◇) it should be ▢. Two weak links meet, forcing opposite colours — a conflict!
    hint: obvious
    technique: GXC
    state:
      selection:
        cells: [R2C1, R3C1, R2C3, R2C9, R7C1, R7C9]

  - text: >
      Rule 3 resolves weak junctions: since R2C9 can't be both colours, it CANNOT contain the digit. Eliminate R2C9~2. The contradiction proves 2 is impossible here.
    hint: detailed
    technique: GXC
    state:
      selection:
        cells: [R2C9]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  coordinateFormat: rncn
  navigation: numbered

Without the grouped node [R2C1, R3C1], this pattern couldn't form — the individual cells R2C1 and R3C1 don't create the necessary strong link structure.

Strong Junction (Rule 2)

Strong Junction occurs when two strong links meet at a node. When this happens, the junction must be TRUE — place the digit there.

Pattern: The chain follows the sequence:

... ═══ junction ═══ ...

Both links entering the junction are strong, meaning if either adjacent node is FALSE, the junction must be TRUE. Since at least one adjacent node must be FALSE (they can't both be TRUE in the chain), the junction is proven TRUE.

Note: Strong Junction patterns with grouped nodes are rare in practice. The principle works identically to X-Cycles Strong Junction — when two strong links meet at a node, that node must contain the digit. The only difference is that one or more links may connect to grouped nodes instead of individual cells.

When you encounter a Strong Junction with grouped nodes:

Identify the junction cell (where two strong links meet)
The junction cell MUST contain the digit
All other candidates in that cell can be eliminated

Why Use Grouped Nodes?

Grouped nodes unlock patterns that standard X-Cycles can't find:

More connections — Grouped nodes can form strong links where individual cells can't. A pointing pair acts as a single unit that "owns" the digit for its line.
Wider reach — A group of cells sees more of the grid than any individual cell, enabling eliminations in more locations.
Breaking stalemates — When standard X-Cycles fail to find eliminations, grouped versions often succeed because they reveal hidden structure in the puzzle.

Complexity

Grouped X-Cycles are expert-level because:

Must identify valid grouped nodes (2-3 cells aligned within a box)
Track both individual and grouped nodes in the same chain
Verify that groups function correctly as units (all cells share the same colour)
Same complex rules as standard X-Cycles

Relationship to Other Techniques

Technique	Node Types	Links
Simple Colouring	Individual only	Strong only
X-Cycles	Individual only	Strong + Weak
Grouped X-Cycles	Individual + Grouped	Strong + Weak
3D Medusa	Individual (multi-digit)	Strong + Bi-value

Grouped X-Cycles is the most powerful single-digit chaining technique before moving to multi-digit chains like 3D Medusa.

Tips

Find pointing patterns first — Look for 2-3 cells with the same candidate aligned in a box. These are your potential grouped nodes.
Try standard X-Cycles first — Only use grouping when individual cells don't form a valid pattern.
Check strong links carefully — A group forms a strong link when the only candidates for that digit in a unit are the group plus one other cell.
Groups share a colour — All cells in a grouped node are assigned the same colour. If the group is TRUE, exactly one cell in it will contain the digit.
Trust the hint system — Grouped chains are difficult to spot manually. The app's chain visualiser shows grouped nodes as connected units.
Watch for multiple groups — A single chain can include several grouped nodes (as in the Nice Loop example above).

Visual Indicators

In Lazy Sudoku's graph display:

Grouped nodes appear as connected cells sharing the same colour
Strong links to/from groups connect to ALL cells in the group
Weak links work similarly — if a chain enters a group, it enters all cells

More Puzzles

Related Techniques

X-Cycles — Standard version without grouping
Pointing Pair/Triple — Creates potential grouped nodes
Simple Colouring — Strong links only
3D Medusa — Multi-digit alternative