Simple Coloring is an advanced Sudoku technique that uses the concept of conjugate pairs to build chains of logical implications for a single candidate digit. A conjugate pair exists when a candidate appears in exactly two cells within a row, column, or box — meaning one of those two cells must contain that digit. By assigning alternating colors (say blue and green) to cells connected through conjugate pairs, you build a chain where all blue cells either all contain the digit, or none of them do (and vice versa for green).
The technique produces eliminations through two main rules. First, if two cells of the same color can see each other (they share a row, column, or box), then that color is invalid — all cells of that color cannot contain the digit, so the opposite color must be correct. This is called a "color trap" or "color contradiction." Second, if an uncolored cell can see cells of both colors, that uncolored cell cannot contain the digit regardless of which color is correct — this is called a "color wrap" or "multi-color elimination."
Simple Coloring is particularly elegant because it deals with only one candidate at a time, making it conceptually simpler than multi-digit chain techniques. It is the gateway to more complex coloring methods like multi-coloring and 3D Medusa. The technique excels in situations where a candidate digit forms long chains of conjugate pairs across the grid, and it can produce eliminations that fish patterns and wing techniques cannot find.
Try It Yourself
Walk through each step of the simple coloring technique on a real puzzle. Follow the instructions and try entering the correct value when prompted.
We will apply Simple Coloring to candidate 4. First, identify conjugate pairs for digit 4. In row 1, 4 appears as a candidate in exactly two cells: (0,4) and (0,7). This is our starting conjugate pair.
Step-by-Step Guide
Choose a candidate digit and identify all conjugate pairs in every row, column, and box.
Pick a starting cell from any conjugate pair and assign it Color A (e.g., blue).
Assign Color B (e.g., green) to the other cell in the conjugate pair.
Extend the chain: from each newly colored cell, find its conjugate partner in other units and assign the alternating color.
Continue until no more conjugate connections can be made.
Check for contradictions: if two cells of the same color see each other, that color is false — the other color is the solution.
Check for eliminations: any uncolored cell that sees both a Color A and a Color B cell can have the candidate removed.
Picture a string of light bulbs wired so that every other one is on — if you know any single bulb's state, you know them all, and anything near both an on and off bulb is guaranteed to be in the dark.
A conjugate pair is a strict binary constraint: within that unit, exactly one of the two cells holds the digit, so assigning true to one forces false on the other. Chaining these binary constraints propagates truth values deterministically, partitioning all connected cells into two mutually exclusive groups. Any external cell that sees members of both groups is eliminated because exactly one group is entirely true, guaranteeing the digit appears in at least one of the two visible cells.
When to use: Use Simple Coloring when a single candidate forms multiple conjugate pairs that can be chained together across the grid. It is especially effective when fish patterns and wing techniques have been exhausted.
Common Mistakes to Avoid
Extending the chain through non-conjugate links where the candidate appears in more than two cells in a unit.
Every link in a Simple Coloring chain must be a conjugate pair — the candidate must appear in exactly two cells in that row, column, or box.
Confusing which rule to apply: contradiction (same color sees itself) versus elimination (uncolored sees both colors).
Contradiction means one entire color group is false. Elimination means a specific uncolored cell loses the candidate. Check both rules separately.
More Examples
See simple coloring applied in different puzzle configurations to strengthen your pattern recognition.
Simple Coloring on Digit 3
Highlighted cells show the simple coloring pattern
Practice Puzzles
Apply the simple coloring technique on these mini challenges. Tap a highlighted cell and enter the correct digit.
Quick Reference
- Pattern:
- A candidate has conjugate pairs forming a chain that can be colored in two colors
- Action:
- Eliminate based on contradiction (same color sees itself) or double-color elimination
- Look for:
- Conjugate pairs of a single candidate that form chains across the grid