Lesson:Coloring

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Coloring involves tracing bi-value links with alternating colors (e.g., Blue and Yellow).

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About This Lesson

Coloring (also called Simple Coloring or Single Digit Coloring) is an elegant and powerful advanced Sudoku technique used to resolve the placement of a specific candidate digit by analyzing the network of cells it could potentially occupy. The technique works by assigning alternating "colors" — commonly blue and yellow, or TRUE and FALSE — to connected cells that share strong links, and then using logical deductions about the coloring to make eliminations or direct placements.

The core insight behind Coloring is that a strong link exists between two cells when exactly two cells in a unit can hold a particular digit. In a strong link, one cell must contain the digit and the other must not — there is no ambiguity. By chaining these strong links and alternating colors, you build a bipartite graph where one color set must be TRUE (containing the digit) and the other must be FALSE.

Coloring is the gateway to more advanced chain-based techniques like X-Cycles and Alternating Inference Chains (AICs). Mastering it builds the intuition for how logical chains work in Sudoku, which becomes critical when solving Diabolical and beyond-rated puzzles.

How It Works — Step by Step

Step 1 — Choose a digit and find all strong links

Select one candidate digit. A strong link (conjugate pair) exists in any unit where the digit appears in exactly two cells. List all conjugate pairs for that digit across all rows, columns, and boxes. These pairs are the edges of your coloring graph.

Step 2 — Paint connected cells with alternating colors

Start at any cell that belongs to a conjugate pair and color it Blue. Color its partner in that conjugate pair Yellow. Now check whether the Blue cell is part of another conjugate pair — if so, the partner in that new pair gets Yellow. Continue this alternating process until the chain can be extended no further.

Step 3 — Look for color conflicts (Rule 1 — Color Traps)

If two cells of the same color sit in the same unit (same row, column, or box), that color cannot be TRUE — it would require two cells in the same unit to hold the digit, which is illegal. Therefore, all cells of that conflicting color are FALSE, and all cells of the opposite color are TRUE. Place the digit in every TRUE-colored cell.

Step 4 — Look for cells seeing both colors (Rule 2 — Color Wings)

If an uncolored cell can see (shares a unit with) a Blue cell and also a Yellow cell, it cannot contain the digit — regardless of which color turns out to be TRUE. Eliminate the digit from that cell. Repeat this check for every uncolored candidate cell.

When to Use This Technique

Use Coloring when X-Wing and simpler pair-based techniques are exhausted, and a particular digit still has many candidate cells spread across the board. Coloring shines when a digit forms long conjugate-pair chains. It is most effective on Hard and Diabolical puzzles. Always try X-Wing first for a given digit, since X-Wing is simpler and is actually a special case of Coloring.

Worked Examples

Suppose digit 5 forms three conjugate pairs: Row 1 (columns 3 and 7), Column 7 (rows 1 and 6), and Row 6 (columns 2 and 7). Painting: R1C3=Blue, R1C7=Yellow. Yellow is connected via column 7 to R6C7=Blue. Blue R6C7 is connected via row 6 to R6C2=Yellow. Now check: is any uncolored cell containing digit 5 able to see both a Blue and a Yellow cell? If R4C2 sees Yellow R6C2 (via column 2) and Blue R4... this elimination collapses a chain of placements.

A simpler Coloring deduction: two same-colored cells in the same box means that color is impossible. Say R2C1 and R5C1 are both Blue — two Blue cells in Column 1. Since the digit cannot appear twice in column 1, Blue must be FALSE. All Blue cells have their candidate digit eliminated; all Yellow cells have the digit placed.

Frequently Asked Questions

Is Coloring the same as X-Wing?

X-Wing is a special case of Coloring involving exactly two rows and two columns forming a rectangle. Coloring is more general — it can handle arbitrarily long chains of conjugate pairs across many units. All X-Wings can be explained by Coloring, but not all Coloring patterns are X-Wings.

Can I use Coloring for multiple digits at once?

Basic Coloring (Single Digit Coloring) focuses on one digit at a time. Multi-digit chain techniques like AICs can involve multiple digits in the chain, but those are separate, more advanced techniques.

What happens if I cannot extend a coloring chain?

A short chain that cannot be extended may still yield Color Wing eliminations (cells seeing both colors). If neither Rule 1 nor Rule 2 produces results, the chain is inconclusive for that digit — try a different digit or a different technique.

How does Coloring relate to Alternating Inference Chains?

Coloring is a subset of AICs restricted to strong links only and a single digit. AICs generalize this to combine strong and weak links and can involve multiple digits. Once you understand Coloring fluently, AICs become a natural extension.

Ready to test your knowledge? Try applying this technique in our Expert Sudoku puzzles, Advanced Sudoku lessons or explore X-Wing technique guide. Keep training to improve your solve times and master the grid!

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