Simple Coloring
Simple Coloring is a powerful expert-level technique that introduces visual chain logic to Sudoku solving. By marking candidates with two alternating colors along chains of strong links, you can track complex relationships across the entire grid and identify eliminations that would be nearly impossible to spot through traditional pattern recognition.
This technique serves as a gateway to advanced chaining methods, teaching you to think in terms of "if this, then that" logic while maintaining visual clarity through color coding. Once you master Simple Coloring, you'll unlock a new dimension of Sudoku solving that makes expert-level puzzles feel manageable.
What is Simple Coloring?
Simple Coloring is a technique that uses two colors (often visualized as blue and green, or Color A and Color B) to mark candidates connected by strong links. A strong link exists when a candidate appears in exactly two cells within a unit—meaning one of those cells must contain the candidate.
Core Concepts
Strong Links (Conjugate Pairs)
A strong link is formed when a candidate appears in exactly two cells in a row, column, or box. For example:
- If candidate 5 appears only in cells R1C2 and R1C7 in Row 1, these cells form a strong link
- One of these cells must be 5 (though we don't know which one yet)
- If one is false, the other must be true—they're complementary
Color Chains
When strong links connect through shared cells, they form chains:
- Start with any strong link and mark one cell blue, the other green
- If the green cell forms another strong link, mark its partner blue
- Continue alternating colors along connected strong links
- The result is a color chain where exactly one color must be true throughout
How Simple Coloring Works
Once you've built color chains, you can make eliminations in two ways:
Rule 1: Color Contradiction
If two cells of the same color can see each other (share a row, column, or box), that color must be false. Why? Because both cells can't be the candidate simultaneously, yet they're the same color (meaning both would be true or both false). Therefore, the opposite color must be true.
Example: If blue cells R1C3 and R1C8 are both candidate 7, they're in the same row and can see each other. Both can't be 7, so blue must be false, meaning all green cells in this chain are 7.
Rule 2: Color Elimination
If a non-colored cell with the candidate can see all instances of one color, you can eliminate that candidate from the non-colored cell. Why? If the colored cells were true, they'd eliminate the non-colored cell. But one color must be true, so if the non-colored cell sees all of one color, it can't contain the candidate.
Example: If R5C5 (not colored) with candidate 7 can see three blue cells (R5C2, R3C5, R6C6), and any one of those blue cells would eliminate R5C5, then you can eliminate 7 from R5C5.
Simple Coloring Example
Scenario: We're tracking candidate 4 and building a color chain.
Step 1: Identify Strong Links
- Row 2: Candidate 4 appears only in R2C3 and R2C8 (strong link)
- Column 3: Candidate 4 appears only in R2C3 and R7C3 (strong link)
- Row 7: Candidate 4 appears only in R7C3 and R7C9 (strong link)
- Box 9: Candidate 4 appears only in R7C9 and R9C8 (strong link)
Step 2: Build the Color Chain
- Start: Mark R2C3 as BLUE
- R2C3 and R2C8 are a strong link → mark R2C8 as GREEN
- R2C3 and R7C3 are a strong link → mark R7C3 as GREEN
- R7C3 and R7C9 are a strong link → mark R7C9 as BLUE
- R7C9 and R9C8 are a strong link → mark R9C8 as GREEN
Color Chain Result:
- BLUE cells: R2C3, R7C9
- GREEN cells: R2C8, R7C3, R9C8
Step 3: Look for Eliminations
Rule 1 Check (Color Contradiction): Do any same-color cells see each other? No contradictions found in this example.
Rule 2 Check (Color Elimination): Cell R8C8 contains candidate 4 (not colored). Does it see all instances of one color?
- R8C8 can see R9C8 (GREEN - same column)
- R8C8 can see R7C9 (BLUE - same box)
- R8C8 sees one cell of each color, so no elimination yet
However, if we find that R2C9 contains candidate 4 and can see R2C8 (GREEN), R7C9 (BLUE, same column), we check: Does R2C9 see all cells of one color? If R2C9 can see both R2C8 and R9C8 (both GREEN cells), we can eliminate 4 from R2C9.
Tips for Using Simple Coloring
1. Pick One Candidate at a Time
Simple Coloring works on a single candidate. Choose a candidate that has many conjugate pairs (strong links) across the grid—typically candidates with constrained placements work best.
2. Use Physical or Digital Colors
On paper, use two colored pencils or highlighters. In digital solvers, use color-marking features. Visual distinction is crucial for tracking chains.
3. Build Complete Chains
Start with one strong link and follow all connections before looking for eliminations. Incomplete chains miss opportunities.
4. Check Both Rules Systematically
After building chains, methodically check:
- Do any same-color cells see each other? (Rule 1)
- Do any non-colored cells see all of one color? (Rule 2)
5. Multiple Independent Chains
You might find several disconnected color chains for the same candidate. Each chain uses its own blue/green labeling (or use different color pairs for different chains to avoid confusion).
6. Document Your Chains
Write down or diagram your color chains. Complex chains are easy to lose track of, and documentation helps verify your logic.
7. Combine with Other Techniques
After making eliminations through coloring, return to simpler techniques. Coloring might unlock hidden singles or create new conjugate pairs.
Common Mistakes to Avoid
Coloring Weak Links
Only color conjugate pairs (exactly two instances in a unit). Don't color candidates that appear 3+ times in a unit—these aren't strong links and will lead to incorrect logic.
Mixing Multiple Candidates
Each coloring analysis focuses on one candidate. Don't mix colors across different candidates—this creates confusion and invalid logic.
Incorrect Seeing Relationships
Remember, cells "see" each other only if they share the same row, column, or box. Cells in different units that happen to be near each other on the grid don't see each other.
Forgetting to Check All Color Instances
When applying Rule 2, ensure the non-colored cell sees ALL instances of one color, not just some. Missing one colored cell invalidates the elimination.
Reversing Color Logic
If you find a contradiction for blue, then green is true, not blue. Make sure you're eliminating the correct color after finding contradictions.
Practice Exercises
Exercise 1: Identifying Strong Links
Candidate 7 appears in Row 4 at cells R4C2, R4C5, and R4C9. Is this a strong link?
Show Answer
Answer: No, this is not a strong link. A strong link requires exactly two instances of the candidate in a unit. Since 7 appears three times in Row 4, it's not a conjugate pair and cannot be colored using Simple Coloring.
Exercise 2: Building a Chain
You have these strong links for candidate 3:
- R1C1 and R1C6 (Row 1)
- R1C6 and R9C6 (Column 6)
- R9C6 and R9C9 (Row 9)
If R1C1 is BLUE, what colors are the other cells?
Show Answer
Answer: R1C1 = BLUE, R1C6 = GREEN (strong link with R1C1), R9C6 = BLUE (strong link with R1C6), R9C9 = GREEN (strong link with R9C6). The chain alternates colors: Blue → Green → Blue → Green.
Exercise 3: Finding Eliminations
You've colored candidate 5: BLUE cells are R2C4 and R5C7; GREEN cells are R2C9 and R5C1. Cell R5C4 (not colored) contains candidate 5. Can you make an elimination?
Show Answer
Answer: Yes! R5C4 can see both R5C7 (BLUE, same row) and R5C1 (GREEN, same row). However, for Rule 2, we need to see all instances of ONE color. R5C4 sees one blue and one green, so direct elimination via Rule 2 doesn't apply here. We'd need more information or a different configuration for elimination.
Frequently Asked Questions
What is Simple Coloring in Sudoku?
Simple Coloring is an expert Sudoku technique that uses two colors to mark candidates connected by strong links (conjugate pairs). By tracking these color chains, you can identify contradictions and make eliminations based on which color must be true or false.
What is a strong link in Simple Coloring?
A strong link (or conjugate pair) is when a candidate appears in exactly two cells within a unit (row, column, or box). Since the candidate must be in one of those two cells, they form a strong link—if one is false, the other must be true.
How do I know which color to eliminate in Simple Coloring?
You eliminate based on color contradictions or eliminations: If two cells of the same color see each other, that color must be false (both can't be true). If a non-colored cell sees all instances of one color, you can eliminate that candidate from the non-colored cell.
Is Simple Coloring the same as X-Cycles?
Simple Coloring is a subset of X-Cycles. X-Cycles include both strong and weak links in alternating patterns, while Simple Coloring focuses only on chains of strong links. Simple Coloring is easier to learn and visualize, making it a great introduction to chain-based logic.
When should I use Simple Coloring?
Use Simple Coloring when you're stuck on expert-level puzzles and simpler techniques (singles, pairs, fish patterns) aren't working. It's particularly effective when you have multiple conjugate pairs for a single candidate, allowing you to build long color chains.