Simple Coloring

Simple Coloring uses a visual approach to track candidate relationships. By assigning colors to cells, you can find contradictions or confirmations that lead to eliminations or placements.

The Core Idea

When a candidate appears exactly twice in a unit (row, column, or box), those two cells form a strong link. One must be true, one must be false.

Coloring assigns each cell one of two colors:

Blue: If this cell is true...
Green: ...then this cell is false (and vice versa)

By following chains of strong links and coloring consistently, you can find:

Cells that see both colors (can be eliminated)
A color that contradicts itself (the other color is true)

Building the Color Chain

Step 1: Find Strong Links

For a single digit, find all units where it appears exactly twice. Each pair is a strong link.

Example for digit 5:

Row 2: cells C3 and C7 only — strong link
Column 3: cells R2 and R6 only — strong link
Box 5: cells R4C4 and R6C5 only — strong link

Step 2: Connect the Links

Strong links that share a cell can be connected:

R2C3 ══════ R2C7     (Row 2 strong link)
  ║
  ║ (Column 3 connects)
  ║
R6C3 ══════ R6C5     (Row 6 strong link? Check it.)

If R6C3 and R6C5 are the only 5s in row 6, they form another strong link. Now we have a chain.

Step 3: Assign Colors

Pick a starting cell, assign Blue. Its strong link partner gets Green. Continue:

R2C3 (Blue) ══════ R2C7 (Green)
  ║
R6C3 (Green) ══════ R6C5 (Blue)

Colors alternate along the chain.

Finding Eliminations

Rule 1: Cells Seeing Both Colors

If a cell (not in the chain) can see both a Blue and a Green cell, it cannot contain the digit.

Why?

Either Blue is true or Green is true
The cell sees a true cell either way
It can't have the digit in either scenario

Example:

        Col 3       Col 5       Col 7
Row 2:  Blue                    Green
Row 4:              ???
Row 6:  Green       Blue

Cell R4C5 sees:

R6C5 (Blue) via column 5
Can it see a Green? Check row 4, column 4 box, etc.

If R4C5 sees both Blue and Green through any combination of row/column/box, eliminate 5 from R4C5.

Rule 2: Color Contradiction

If two cells of the SAME color see each other, that color is impossible. The OTHER color is entirely true.

Why?

Same color means both would be true or both false
If they see each other, both can't be true
So that color is false, the other is true

Example:

R2C3 (Blue) ══════ R2C7 (Green)
  ║
R6C3 (Green) ══════ R6C7 (Blue!)

If R2C7 and R6C7 are both Green... wait, let me redo:

R2C3 (Blue) ═══ R2C7 (Green)

R6C3 (Green) ═══ R6C7 (???)

R6C7 connects to R2C7 via column 7. They share a unit.

R2C7 is Green
R6C7 connects to Green, so R6C7 is Blue

But what if:

R2C3 (Blue) ═══ R2C7 (Green) ═══ R6C7 (Blue)
               (via column 7)

And R6C7 also connects to R6C3:

R6C3 (Green) ═══ R6C7 (Blue)

If this is consistent, no contradiction. But if:

R6C3 (Green) ═══ R6C7 (Blue)
                   ↕
                column 7 connects to R2C7 (Green)

Wait, R6C7 via column 7 to R2C7:

R6C7 is Blue, R2C7 is Green — opposite colors. Consistent!

Contradiction example:

What if the chain eventually assigns the same cell two colors?

Path A: R2C3 (Blue) → R2C7 (Green) → R8C7 (Blue)
Path B: R2C3 (Blue) → R5C3 (Green) → R5C7 (Blue) → R8C7 (Green!)

R8C7 is both Blue and Green? Impossible.

If this happens, one color leads to a contradiction. Determine which and eliminate that entire color's cells.

Types of Simple Coloring

Type 1: Color Trap

A cell outside the chain sees both colors → eliminate from that cell.

This is the most common result.

Type 2: Color Wrap

Two cells of the same color see each other → that color is false.

All cells of the false color can have the digit eliminated. All cells of the true color ARE the digit.

Worked Example

Digit 4 strong links:

Row 1: R1C2, R1C8
Row 7: R7C2, R7C6
Column 2: R1C2, R7C2
Column 8: R1C8, R4C8
Box 6: R4C8, R6C9

Build the chain:

Start at R1C2 = Blue

R1C2 (Blue) ═══ R1C8 (Green)     [row 1]
  ║              ║
  ║              ╠══ R4C8 (Blue)  [column 8]
  ║                    ║
  ║                    ╠══ R6C9 (Green)  [box 6]
  ║
  ╠══ R7C2 (Green)     [column 2]
         ║
         ╠══ R7C6 (Blue)  [row 7]

Colors assigned:

Blue: R1C2, R4C8, R7C6
Green: R1C8, R7C2, R6C9

Look for eliminations:

Find cells with candidate 4 that see both Blue and Green.

Cell R7C8:

Sees R7C6 (Blue) via row 7
Sees R4C8 (Blue) via column 8
Sees R1C8 (Green) via column 8

R7C8 sees Blue and Green! Eliminate 4 from R7C8.

Cell R4C2:

Sees R1C2 (Blue) via column 2
Sees R7C2 (Green) via column 2

R4C2 sees both colors! Eliminate 4 from R4C2.

How to Implement Coloring

Manual Method

Choose a digit
Find all strong links
Draw connections on paper
Alternate colors along chains
Search for traps (sees both) or wraps (same color sees itself)

Mental Method

Pick a starting cell, call it "true"
Its partner is "false"
Follow the chain, alternating
Note cells that see both "true" and "false" candidates

Software Assistance

Many Sudoku apps highlight strong links or color candidates automatically. Use these to learn the visual patterns.

Common Mistakes

Mistake 1: Using weak links in the chain

Coloring only works with strong links (exactly 2 candidates in a unit). Cells with 3+ candidates in a unit don't create strong links for coloring.

Mistake 2: Inconsistent coloring

If you color a cell Blue, its strong link partner MUST be Green. Don't accidentally color both the same.

Mistake 3: Missing connections

A cell might connect to the chain through any unit — row, column, or box. Check all three.

Mistake 4: Stopping too early

Keep extending the chain until no more strong links connect. Larger chains have more elimination potential.

Simple Coloring vs. Other Techniques

Technique	Uses	Finds
Simple Coloring	Strong links for one digit	Eliminations for that digit
Multi-Coloring	Multiple color chains	Inter-chain eliminations
X-Chains	Strong links in a chain	Similar eliminations, different notation

Simple Coloring is essentially X-Chains with a visual/color-based approach.

Quick Reference

Strong link: A digit appears exactly twice in a unit.

Coloring rules:

Start with Blue, partner is Green
Alternate along connected strong links
One color is true, one is false

Elimination patterns:

Color Trap: Cell sees both colors → eliminate
Color Wrap: Same color sees itself → that color is false, eliminate all its cells

Finding it:

Pick a digit
Find strong links
Build connected chains
Assign alternating colors
Look for traps or wraps

When to use:

Expert+ puzzles
Digits with clear strong link networks
After simpler techniques fail