Chain Colouring
Several advanced Sudoku techniques use alternating colours to represent logical states along a chain. This article explains the visual system used to display these colours.
The Two-Colour System
In chain-based techniques, candidates are assigned one of two colours to represent opposite logical states:
- If one candidate is true (the solution), all candidates of the opposite colour must be false
- If one candidate is false (eliminated), all candidates of the opposite colour must be true
The key insight: exactly one colour group contains all the true values.
Visual Symbols
To improve accessibility for colour-blind users and greyscale displays (such as e-Ink readers), the visual system uses three independent channels:
- Colour: Blue, Orange, Grey, Red
- Shape: Square, Diamond, Circle, Triangle
- Dash Pattern: Different line styles for links
Vertex Shapes
Each candidate vertex is marked with a shape indicating its colour state:
| State | Shape | Colour | Description |
|---|---|---|---|
| Colour A | Square (▢) | Blue | First colour partition |
| Colour B | Diamond (◇) | Orange | Second colour partition |
| Uncoloured | Circle (○) | Grey | Not yet assigned to a colour |
| Conflict | Triangle (△) | Red | Contradiction detected — both colours apply |
Link Patterns
Links between candidates use dash patterns to reinforce colour distinctions:
| Colour | Pattern | Visual |
|---|---|---|
| Colour A | Dash-dot | ─ · ─ · |
| Colour B | Long dash | ── ── ── |
| Conflict/Shared | Solid | ───── |
| Uncoloured | Dotted | · · · · |
Link Weights
Link strength is shown through line thickness:
| Strength | Weight | Description |
|---|---|---|
| Strong | Thick (3px) | Conjugate pairs — exactly two candidates |
| Weak | Medium (1.5px) | Shared unit but not conjugate |
| Sees | Thin (1px) | Visibility relationship, slightly transparent |
Shape Reference
- Square (Blue): Candidates assigned to the first colour group
- Diamond (Orange): Candidates assigned to the second colour group
- Circle (Grey): Candidates that are part of the chain but haven't been assigned a colour yet
- Triangle (Red): A conflict state where a candidate has been assigned both colours — this indicates a logical contradiction
How Colours Propagate
Colours alternate along strong links:
- Start with any candidate and assign it Square (Blue)
- Its conjugate partner (connected by a strong link) becomes Diamond (Orange)
- Continue alternating: Blue → Orange → Blue → Orange...
This creates two partitions: all Squares and all Diamonds. Since exactly one partition is true, finding a contradiction in one partition proves the other is true.
Colour Inversion
The choice of which endpoint gets Square (Blue) vs Diamond (Orange) is arbitrary — you can start from either end. The app may show colours "inverted" (swapped) compared to other sources, but the logic remains identical.
What matters is the relationship between colours:
- Same shape = same logical state
- Different shape = opposite logical state
Techniques Using Chain Colouring
The following techniques use this alternating colour system:
| Technique | Description |
|---|---|
| Simple Colouring | Single-digit chains using conjugate pairs |
| X-Cycles | Single-digit cycles with strong and weak links |
| 3D Medusa | Multi-digit chains using both bi-location and bi-value links |
| Grouped X-Cycles | X-Cycles using grouped strong links |
Elimination Rules
When a chain has been coloured, eliminations can be found using these rules:
Rule 2: Same Colour Twice in a Unit
If two candidates of the same colour appear in the same unit (row, column, or box), that colour is false:
- Two Squares (Blue) in the same row → All Squares are false → All Diamonds are true
Rule 4: Sees Both Colours
If an uncoloured candidate can see (shares a unit with) candidates of both colours, it can be eliminated:
- The cell sees both a Square and a Diamond
- One of them must be true
- Either way, the uncoloured candidate is false
Tips for Reading Chain Displays
- Focus on shapes, not colours: The shapes (▢ ◇ ○ △) work even without colour
- Look for conflicts: A Triangle (△) immediately signals a contradiction
- Trace the chain: Follow the alternating pattern to understand the logic
- Check for Rule 4: Look for uncoloured cells that see both shapes