Coloring Strategy (Simple Colors)

Coloring tracks a single candidate through chains of conjugate pairs using two alternating colors to identify contradictions and make eliminations.

Difficulty (Chain technique)

Advanced

Targets (One candidate at a time)

Single digit

Colors used (True / false alternation)

2

Required pattern (Pairs of bilocal cells)

Conjugate

What Is Coloring?

Coloring (also known as Simple Coloring or Single's Chains) focuses on a single candidate number. The technique starts by finding conjugate pairs - units where the candidate appears in exactly two cells. These pairs form natural "either/or" relationships: if one cell has the number, the other cannot. By assigning two colors (say Blue and Green) to the cells of each conjugate pair, you build a chain that extends across the grid.

As you follow the chain, colors alternate at each link. If cell A is Blue, its conjugate B in the same row is Green. If B connects to another conjugate C in a column, C is Blue again. The chain grows until all connected conjugate pairs are colored. One color group represents the cells where the candidate IS placed; the other represents where it is NOT.

Two types of eliminations emerge. In a Type 1 contradiction, two cells of the same color see each other - meaning that color cannot be true, so you eliminate the candidate from all cells of that color. In a Type 2 elimination, an uncolored cell sees cells of both colors - since one color must be true, the uncolored cell cannot hold the candidate. For key terms, see our Sudoku glossary.

How to Apply Coloring: Step-by-Step

1. Choose a candidate number - Select a number that appears frequently as a candidate and has multiple conjugate pairs (units where it appears in exactly two cells). Numbers with many conjugate links produce longer, more useful chains.
2. Find a conjugate pair to start - Pick any unit (row, column, or box) where your chosen candidate appears in exactly two cells. Assign Color A (e.g., Blue) to one cell and Color B (e.g., Green) to the other. These are your starting links.
3. Extend the chain - Look at each colored cell and check if it forms a conjugate pair in another unit. If a Blue cell has a conjugate in another row, that conjugate becomes Green. Continue extending, alternating colors at each new link, until no more connections exist.
4. Check for Type 1 contradictions - Examine all cells of each color. If two cells of the same color share a unit (row, column, or box), that color is false. Eliminate the candidate from ALL cells of that color. The other color is confirmed as true.
5. Check for Type 2 eliminations - Look for uncolored cells that contain the candidate. If any such cell shares a unit with a Blue cell AND shares a unit with a Green cell, the candidate can be eliminated from that cell - one of the two colors must be true, blocking the candidate.

When to Use Coloring

Coloring is most effective when the grid has many conjugate pairs for a particular candidate. It excels in situations where fish patterns like X-Wing and Swordfish are not applicable, and XY-Wing patterns are absent.

This technique shines in hard and expert puzzles that have resisted other advanced methods. It is also a stepping stone toward understanding more complex chain-based techniques like Forcing Chains. For the full technique progression, see our strategy guide or learn how to play Sudoku.

Difficulty Level & Puzzle Types

Coloring is classified as an advanced-level technique. It requires understanding conjugate pairs, chain construction, and visibility analysis - skills that set advanced solvers apart.

Easy

Never needed - basic techniques solve these

Medium

Not needed - intermediate techniques handle these

Hard

Sometimes needed as an alternative to fish patterns

Expert

Frequently useful - a versatile tool for expert solving

Tips for Mastering Coloring

1. Start with candidates that have many conjugate pairs across the grid. Long chains provide more opportunities for contradictions and eliminations.
2. Use pencil marks or a separate notation to track colors. On paper, use two different colored pens or symbols (circle vs. square) to distinguish colors. Final Sudoku includes a built-in coloring tool - tap the Color button in the toolbar or press C to enter color mode, then tap any cell to cycle through four background colors (Blue, Green, Orange, Red). Tap again to clear. Right-click or long-press the Color button to clear all colors at once.
3. Always check for Type 1 contradictions first - they are more powerful because they eliminate the candidate from multiple cells at once and confirm placements.
4. After eliminations, re-scan for Naked Singles and Hidden Singles. Coloring eliminations often cascade into simpler solves.

Frequently Asked Questions

What is the Coloring technique in Sudoku?

Coloring (also called Simple Coloring or Single's Chains) is an advanced technique where you assign two alternating colors to a candidate number along chains of conjugate pairs. A conjugate pair exists when a candidate appears in exactly two cells within a unit. By tracing these chains, you can identify contradictions or confirm placements.

What is a conjugate pair in Sudoku?

A conjugate pair occurs when a specific candidate number appears in exactly two cells within a row, column, or box. These two cells are "conjugates" - if one contains the number, the other cannot, and vice versa. Conjugate pairs form the links in a coloring chain.

How do the two colors work in Sudoku Coloring?

You assign Color A to one cell of a conjugate pair and Color B to the other. As you extend the chain through connected conjugate pairs, colors alternate. If a cell is Color A, its conjugate in the same unit gets Color B, and so on. Eventually, one color group represents cells that DO contain the candidate, and the other group represents cells that do NOT.

What types of eliminations can Coloring produce?

There are two main elimination types. Type 1 (Contradiction): If two cells of the same color see each other, that color is false - eliminate the candidate from all cells of that color. Type 2 (Seeing both colors): If any uncolored cell sees cells of both colors, the candidate can be eliminated from that cell because one of the two colors must be true.

When should I use Coloring instead of other advanced techniques?

Use Coloring when fish patterns (X-Wing, Swordfish) and XY-Wings have not resolved the puzzle. Coloring is especially effective when a candidate forms long chains of conjugate pairs across the grid. It is a versatile alternative that can sometimes find eliminations that fish patterns miss.

Related Techniques

• XY-Wing vs Coloring - Compare lighter chain technique
• Forcing Chains vs Coloring - Heavy chain comparison
• X-Wing - Rectangular fish pattern for candidate elimination
• Forcing Chains - Follow chains of implications to derive eliminations
• XY-Wing - Three bi-value cells forming an elimination chain
• Unique Rectangles - Use uniqueness to prevent multiple solutions

Related Glossary Terms

• Candidate
• Cell
• Elimination
• Singles
• Row & Column

View full glossary

Practice by Difficulty

• Easy Sudoku - Solving guide
• Medium Sudoku - Solving guide
• Hard Sudoku - Solving guide
• Expert Sudoku - Solving guide

See it in action: Simple Coloring on digit 5

5

5

5

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Track digit 5. We've found three 'conjugate pairs' — units where 5 fits in only two cells. Let's color the cells: A and B alternating along the chain.

Step 1 of 4 — full interactive demo available with JavaScript enabled