Simple Colouring

Simple Colouring (also called Singles Chains) uses conjugate pairs to build a chain of alternating colours for a single digit. By analysing the colour pattern, we can find eliminations or solutions.

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

How It Works

Building the Chain

For a chosen digit:

  1. Find all strong links (cells where the digit appears in exactly two places in a unit)
  2. Assign colours: if one cell is Square (Blue), its conjugate partner is Diamond (Orange)
  3. Extend the chain: when a coloured cell has another strong link, colour its partner with the opposite colour
  4. Continue until no more connections

The Logic

In a properly coloured chain:

How to Spot Simple Colouring

Conjugate Pair Focus

The key to spotting Simple Colouring is finding conjugate pairs — units where a digit appears in exactly two cells:

Step Action What to Look For
1 Choose a digit Pick a digit with multiple conjugate pairs across the grid
2 Find conjugate pairs Units (row, column, box) where digit appears in exactly 2 cells
3 Build the chain Alternate colours along strong links (Square → Diamond → Square)
4 Check for contradictions Same colour appears twice in a unit (Rule 2)
5 Check for eliminations Uncoloured cell sees both colours (Rule 4)

Step-by-Step Scanning

  1. Identify strong link candidates — Look for digits with good conjugate pair coverage
  2. Start colouring — Pick any conjugate pair and assign Square (Blue)/Diamond (Orange)
  3. Extend the chain — Follow conjugate pairs, alternating colours
  4. Look for Rule 2 — Same colour twice in a unit = contradiction
  5. Look for Rule 4 — Uncoloured cell seeing both colours = elimination

Visualisation Tip

The app displays chains with:

Example

This example demonstrates Rule 2 (Twice in Unit) — finding a contradiction where the same colour appears twice in one unit.

Pattern Analysis:

puzzle: S9B0L2D04080E2D030I0F7R059F2B2F060H04020O080F0D090O2Q2B0Z069H0H0E2D047O0N7R7P039F0F089F7W057V0D7P0E030L7N0F0H070E7N7R9F065Z9M02BE07047R020N478206BM0H0F029E0D05012E7Q
mode: guided
technique: Simple Colouring
initial:
  layers:
    hints: true
steps:
  - text: >
      Simple Colouring builds chains from conjugate pairs — units where a digit appears in exactly two cells. Look for digit 7.
    hint: subtle
    technique: SC

  - text: >
      Use the links display to see conjugate pairs. Solid lines connect cells that form strong links for digit 7.
    hint: subtle
    technique: SC
    state:
      graph:
        type: strongChain
        digits: [7]

  - text: >
      R1C2 and R2C3 form a conjugate pair in box 1 — if one is 7, the other cannot be. Assign them opposite colours (Square/Diamond).
    hint: subtle
    technique: SC
    state:
      selection:
        cells: [R1C2, R2C3]
      graph:
        type: strongChain
        digits: [7]

  - text: >
      Extend the chain: R1C2 links to R4C2 (column 2) and R1C6 (row 1). R2C3 links to R5C3 (column 3). Each link alternates colour.
    hint: subtle
    technique: SC
    state:
      selection:
        cells: [R1C2, R1C6, R2C3, R4C2, R5C3]
      graph:
        type: strongChain
        digits: [7]

  - text: >
      Continue building: R4C2 links to R4C5, and R5C3 links to R5C6. The full chain has 8 cells with alternating Square/Diamond colours.
    hint: obvious
    technique: SC
    state:
      selection:
        cells: [R1C2, R1C6, R2C3, R2C5, R4C2, R4C5, R5C3, R5C6]

  - text: >
      Rule 2 check: Look for same colour appearing twice in a unit. R4C2 and R5C6 are BOTH Diamond (Orange) — and they share box 4!
    hint: detailed
    technique: SC
    state:
      selection:
        cells: [R4C2, R5C6]

  - text: >
      Contradiction! Diamond (Orange) cannot be true (would put two 7s in box 4). Eliminate 7 from ALL Diamond cells: R1C6, R2C3, R2C5, R4C2, R5C6.
    hint: detailed
    technique: SC
    state:
      selection:
        cells: [R1C6, R2C3, R2C5, R4C2, R5C6]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered

Elimination Rules

Rule 2: Twice in a Unit

If the same colour appears twice in the same row, column, or box:

The main example above demonstrates Rule 2 — Diamond (Orange) appears twice in box 4, proving Diamond is false.

Rule 4: Two Colours See a Cell

If an uncoloured cell sees BOTH a Square (Blue) cell AND a Diamond (Orange) cell:

Pattern Analysis:

puzzle: S9B029Y6E5Y0D0ACY12060D9Y6E066E02CY0A5Y5U010F5Y092U5W1404032Q6A0A0B09060D5U0A040B5Y0F2E0E095Y5U0F09055Y0D5W0O01050H0D0B010F03070I0I0B2E042E080A0F0E062E0A0I2U2U0D0H02
mode: guided
technique: Simple Colouring
initial:
  layers:
    hints: true
steps:
  - text: >
      Rule 4 finds eliminations without contradictions. Look for digit 3 — where does it appear in exactly two cells per unit?
    hint: subtle
    technique: SC

  - text: >
      Enable links display to see the conjugate pairs for digit 3. Solid lines show strong links between cells.
    hint: subtle
    technique: SC
    state:
      graph:
        type: strongChain
        digits: [3]

  - text: >
      R2C9 and R5C9 form a conjugate pair in column 9. Assign R2C9 Square (Blue), R5C9 Diamond (Orange).
    hint: subtle
    technique: SC
    state:
      selection:
        cells: [R2C9, R5C9]
      graph:
        type: strongChain
        digits: [3]

  - text: >
      Extend: R5C9 links to R6C8 (box 6). R6C8 becomes Square (Blue). R6C8 links to R6C5. R6C5 becomes Diamond (Orange).
    hint: subtle
    technique: SC
    state:
      selection:
        cells: [R2C9, R5C9, R6C5, R6C8]
      graph:
        type: strongChain
        digits: [3]

  - text: >
      Now check Rule 4: R2C5 has candidate 3 but is NOT in the chain. Does it see both colours?
    hint: obvious
    technique: SC
    state:
      selection:
        cells: [R2C5]

  - text: >
      R2C5 sees R2C9 (Square, same row) AND R6C5 (Diamond, same column). One of them MUST be 3, so R2C5 cannot be 3. Eliminate R2C5~3.
    hint: detailed
    technique: SC
    state:
      selection:
        cells: [R2C5, R2C9, R6C5]
settings:
  showCandidates: true
  showControls: true
  showDescription: true
  navigation: numbered

Tips

  1. Pick the right digit — Choose a digit with good strong link coverage
  2. Build carefully — Only extend through genuine conjugate pairs
  3. Check all rules — Rule 4 is most common, but Rule 2 gives more eliminations
  4. Separate chains — Multiple disconnected chains may exist for the same digit

When to Use

Simple Colouring is typically useful when:

Chain Visualisation

The app can display colouring chains:

More Puzzles

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