Multi-Coloring
Multi-Coloring extends Simple Coloring into a more sophisticated realm, analyzing how multiple independent color chains for the same candidate interact with each other. While Simple Coloring examines one chain in isolation, Multi-Coloring reveals hidden relationships between separate chains, unlocking eliminations that would otherwise remain invisible.
This expert-level technique represents a significant leap in complexity, requiring you to track multiple color pairs simultaneously and understand how chains constrain each other. Mastering Multi-Coloring places you among the elite of Sudoku solvers, equipped to tackle the most diabolical puzzles with pure logic.
What is Multi-Coloring?
Multi-Coloring (also called Medusa or 3D Coloring) is a technique that identifies and analyzes multiple independent color chains for a single candidate, then examines how these chains interact to create eliminations.
Building Multiple Chains
Just as in Simple Coloring, you build chains using strong links (conjugate pairs). However, you'll often find that a candidate has several disconnected chains:
- Chain 1: Color with Blue/Green (or colors A/A')
- Chain 2: Color with Red/Yellow (or colors B/B')
- Chain 3: Color with Orange/Purple (or colors C/C')
Each chain maintains its own two-color system where exactly one color in each chain must be true.
Key Multi-Coloring Rules
Rule 1: Color Contradiction Between Chains
If two cells of specific colors from different chains can see each other, it creates constraints:
- Type 1: If Blue (Chain 1) and Red (Chain 2) see each other, then both cannot be true simultaneously. Therefore, at least one of their complementary colors (Green or Yellow) must be true.
- Type 2: If Blue and Red see each other AND Green and Yellow also see each other, then you've found a contradiction that determines which colors are true.
Rule 2: Color-on-Color Elimination
If a cell from one chain can see cells from another chain in a specific pattern:
- If Blue (Chain 1) sees both Red and Yellow (Chain 2's both colors), then Blue must be false (because no matter which Chain 2 color is true, Blue is eliminated).
Rule 3: Non-Colored Cell Elimination
If a non-colored cell with the candidate can see specific color combinations across multiple chains, you can eliminate the candidate from that cell.
The Logic Behind Multi-Coloring
Multi-Coloring works because:
- Within each chain, exactly one color must be true
- All chains represent the same candidate
- Only one instance of the candidate can be true in any unit
- When chains interact through seeing relationships, they constrain each other's possible truth values
Multi-Coloring Example
Scenario: We're tracking candidate 6 and have found two independent color chains.
Chain 1 (Blue/Green):
- Blue: R1C4, R7C6
- Green: R1C9, R7C2
Chain 2 (Red/Yellow):
- Red: R3C2, R5C9
- Yellow: R3C7, R5C4
Analysis of Chain Interactions:
Checking for Seeing Relationships:
- R7C2 (Green, Chain 1) and R3C2 (Red, Chain 2) share Column 2 → they see each other
- R1C9 (Green, Chain 1) and R5C9 (Red, Chain 2) share Column 9 → they see each other
What This Means:
Green and Red see each other in multiple locations (R7C2-R3C2 and R1C9-R5C9). This means Green and Red cannot both be true. Therefore:
- If Green is true, Red must be false (and Yellow must be true)
- If Red is true, Green must be false (and Blue must be true)
Finding Eliminations:
Now we look for cells that would be eliminated in BOTH scenarios:
Consider cell R4C4 (not colored) contains candidate 6. Check what it sees:
- R4C4 sees R1C4 (Blue, Chain 1 - same column)
- R4C4 sees R5C4 (Yellow, Chain 2 - same column)
Applying Logic:
- Scenario A: If Green is true, then Blue is false, so R1C4 is not 6. But Yellow must be true, so R5C4 is 6, eliminating R4C4.
- Scenario B: If Red is true, then Yellow is false, so R5C4 is not 6. But Blue must be true, so R1C4 is 6, eliminating R4C4.
In BOTH scenarios, R4C4 cannot be 6! We can eliminate candidate 6 from R4C4.
Tips for Using Multi-Coloring
1. Master Simple Coloring First
Multi-Coloring is significantly more complex. Don't attempt it until you're comfortable building and analyzing single color chains with Simple Coloring.
2. Use Distinct Color Pairs
Assign clearly different color pairs to each chain:
- Chain 1: Blue/Green
- Chain 2: Red/Yellow
- Chain 3: Orange/Purple
This visual distinction prevents confusion when analyzing interactions.
3. Build All Chains Before Analyzing
Complete all color chains for the candidate before looking for interactions. Incomplete chains miss relationships.
4. Systematically Check Chain Pairs
For each pair of chains, check if colors see each other:
- Does Blue see Red?
- Does Blue see Yellow?
- Does Green see Red?
- Does Green see Yellow?
5. Document Chain Relationships
Write down which colors see each other. For complex puzzles, create a diagram showing chain interactions. This documentation is essential for avoiding errors.
6. Focus on Constrained Candidates
Multi-Coloring works best on candidates that already have many conjugate pairs. Highly constrained candidates are more likely to form multiple useful chains.
7. Verify Eliminations Carefully
Multi-Coloring logic is complex and error-prone. Before eliminating a candidate, trace through the logic multiple times to ensure correctness.
Common Mistakes to Avoid
Mixing Chain Colors
Never mix colors between chains. Each chain has its own independent color pair. Blue from Chain 1 is completely separate from Blue in Chain 2 (if you reuse colors).
Incomplete Chain Building
Building partial chains misses critical interactions. Always extend chains as far as possible before analyzing.
Forgetting Complementary Colors
When Blue and Red see each other, remember that this also constrains Green and Yellow. Always consider both colors in each chain when analyzing interactions.
Incorrect Seeing Relationships
Two cells see each other only if they share a row, column, or box. Double-check all seeing relationships, as errors here invalidate all subsequent logic.
Over-Complicating Simple Situations
If Simple Coloring finds eliminations, use it! Don't unnecessarily escalate to Multi-Coloring unless single chains fail to progress.
Assuming Connected Chains
Independent chains are not connected by strong links—they're separate networks. Don't assume color relationships between chains unless proven through seeing relationships.
Practice Exercises
Exercise 1: Understanding Chain Independence
You have two chains for candidate 8: Chain 1 (Blue/Green) and Chain 2 (Red/Yellow). If you determine Blue must be true, does this tell you anything about Red or Yellow?
Show Answer
Answer: Not by itself. Without seeing relationships between the chains, determining Blue is true only tells you that Green is false within Chain 1. The chains are independent until you find cells of different colors that see each other, creating interactions.
Exercise 2: Chain Interaction
Chain 1: Blue = R2C3, Green = R2C8. Chain 2: Red = R2C5, Yellow = R9C5. What interaction exists?
Show Answer
Answer: Blue (R2C3), Red (R2C5), and Green (R2C8) all share Row 2, so they all see each other. This means Blue, Red, and Green cannot all be true simultaneously. Since exactly one color per chain must be true, this creates complex constraints: at most one of {Blue, Green} and at most one of {Red, Yellow} can be true, but they also can't all be in Row 2.
Exercise 3: Elimination Logic
You've determined that Blue (Chain 1) and Red (Chain 2) cannot both be true. Cell R5C5 (not colored) sees Blue at R5C2 and Yellow at R8C5. Can you eliminate candidate from R5C5?
Show Answer
Answer: No, you cannot eliminate based on this information alone. R5C5 sees Blue and Yellow, but these are complementary (opposites) in the constraint. If Blue is false, Red is true (so Yellow is false), meaning R5C5 only sees one true color. If Blue is true, Yellow might be false, but we need to see both colors of one chain to guarantee elimination.
Frequently Asked Questions
What is Multi-Coloring in Sudoku?
Multi-Coloring is an expert Sudoku technique that analyzes interactions between multiple independent color chains for the same candidate. By examining how different color chains relate to each other, you can find eliminations that single-chain Simple Coloring cannot detect.
How is Multi-Coloring different from Simple Coloring?
Simple Coloring analyzes one color chain at a time using two colors. Multi-Coloring examines multiple independent chains simultaneously, using multiple color pairs (like Blue/Green for chain 1, Red/Yellow for chain 2). It finds eliminations based on how these separate chains interact.
What are the main Multi-Coloring elimination rules?
Multi-Coloring has two main elimination rules: 1) If colors from different chains see each other in specific combinations, one or both chains have determined colors. 2) If a non-colored cell sees specific color combinations across multiple chains, you can eliminate that candidate from the cell.
Is Multi-Coloring harder than Simple Coloring?
Yes, Multi-Coloring is significantly more complex than Simple Coloring. It requires tracking multiple color chains simultaneously and understanding how they interact. You should master Simple Coloring thoroughly before attempting Multi-Coloring.
When should I use Multi-Coloring?
Use Multi-Coloring when Simple Coloring doesn't yield eliminations but you have multiple independent color chains for the same candidate. It's an expert-level technique for the most difficult puzzles, typically used after exhausting simpler methods.