Coloring in Sudoku: Simple and Advanced Methods
Coloring in Sudoku is a candidate elimination technique that works by assigning two colors (like 'A' and 'B') to alternating instances of a single candidate digit along strong links. The method reveals contradictions—like two cells of the same color ending up in the same row, column, or box—which allows you to eliminate all candidates of that color, or sometimes confirm the opposite color as correct. This powerful strategy, which includes both Simple Coloring and the more advanced 3D Medusa, helps solve puzzles where basic methods and even other advanced patterns like the [X-Wing](/techniques/x-wing) or [Y-Wing](/techniques/y-wing) fail to provide progress.
The Core Principle: Strong Links and Alternating Colors
At its heart, coloring is based on tracking a binary possibility. You start with a strong link, which exists when a specific candidate number appears exactly twice in a row, column, or 3x3 box. These two cells form a 'this-or-that' relationship: the digit must be in one of them. By coloring one cell in the pair as Color A, you logically force the other to be Color B. You then extend this chain by finding other strong links for the same candidate number connected to your colored cells. Each new link forces an alternating color, creating a chain of As and Bs that maps out all possible placements for that digit across the grid.
Simple Coloring: One Candidate, Two Colors
Simple Coloring applies this principle to a single candidate digit (e.g., all the possible 5s). You systematically color the entire chain for that number. The process is: 1) Pick a candidate that appears in several strong links. 2) Choose a starting strong link and label its two cells A and B. 3) For every colored cell, look in its row, column, and box. If you find another cell with the same candidate that forms a strong link with it, color that new cell with the opposite color. You continue this propagation until no more strong links for that candidate can be added to the chain.
- Use simple, clear notation like marking cells with a small 'A' or 'B' in the corner, or using two distinct highlight colors if solving on an app.
- If a cell in the chain is already solved or the candidate is eliminated, the chain stops there. You only color cells where the candidate is still possible.
How Contradictions Reveal Eliminations
The power of coloring comes from finding contradictions within the colored chain. There are two primary types of contradictions that allow for eliminations. First, if two cells of the same color (e.g., both Color A) ever appear in the same row, column, or 3x3 box, you have a direct conflict. This means that color cannot possibly be the solution, as it would place the same digit twice in one unit. Therefore, you can eliminate the candidate from all cells marked with that losing color. Second, if an uncolored cell 'sees' (is in the same row, column, or box as) cells of both colors A and B, that uncolored cell can never contain the candidate. One of the colored cells must be true, so the candidate is already accounted for in that unit.
Advanced Coloring: 3D Medusa and Multi-Candidate Chains
When Simple Coloring for one number doesn't yield a contradiction, you can escalate to 3D Medusa. This advanced technique expands the coloring web to include multiple candidate digits and different types of links. In 3D Medusa, you still use only two colors, but you connect chains from different numbers through bivalue cells (cells with only two candidates). If a cell contains candidates '3' and '7', and the '3' is colored A, then the '7' in that same cell is forced to be color B. This creates a multi-candidate network. Finding a contradiction in this larger, interconnected web—like a cell where two of its own candidates would have to be the same color, or a unit containing the same candidate in the same color from different chains—allows for massive eliminations and is often the key to solving extreme puzzles. For a broader strategy on tackling difficult grids, see our guide on How to Solve Hard Sudoku.
When to Use Coloring in Your Solve
Coloring is a mid-to-late game strategy. Our hint engine data shows players should first exhaust all basic techniques (singles, pairs, triples) and then look for X-Wing, Swordfish, and Y-Wing patterns. When the grid is saturated with candidates and these 'fish' and 'wing' patterns are not apparent, that's the ideal time to look for strong links to build a color chain. Start with the candidate that appears in the most clear strong links across the grid. Simple Coloring is faster to apply and should be tried before constructing a complex 3D Medusa web. Mastering coloring transforms it from a last resort into a reliable tool for breaching the final logical barrier in the hardest puzzles.
- Scan for 'bi-value' cells and 'bi-local' units (where a candidate appears exactly twice) to quickly find strong links to start your chain.
- If coloring yields no immediate contradiction, note the colored grid. It often simplifies the next step, like making an X-Wing pattern visually obvious.
Key Facts
- ▪Sudoku coloring is a candidate elimination technique that assigns two colors (like A and B) to a chain of cells containing the same candidate number.
- ▪The technique relies on strong links, which occur when a candidate appears exactly twice in a row, column, or 3x3 box.
- ▪In Simple Coloring, you propagate colors for a single digit: if one cell in a strong link is color A, the other must be color B.
- ▪A key elimination occurs if two cells of the same color end up in the same row, column, or box, proving that color is impossible.
- ▪You can also eliminate a candidate from any uncolored cell that 'sees' (shares a unit with) both colors A and B of that candidate.
- ▪3D Medusa is an advanced form of coloring that builds chains using multiple candidate numbers connected through bivalue cells.
- ▪Coloring is typically used after simpler advanced techniques like X-Wing and Y-Wing fail to provide progress.
- ▪The method does not identify the correct cell directly but proves an entire set of possibilities (one color) is false.
Frequently Asked Questions
What is the difference between Simple Coloring and 3D Medusa?
Simple Coloring uses one candidate number. 3D Medusa is more advanced, linking chains of different candidates through bivalue cells using the same two-color logic, creating a larger network for finding contradictions.
Do I need to use actual colors on paper?
No. You can use letters (A/B), symbols (●/○), or two different pencil marks. The concept is binary logic, not visual color.
When should I look for a coloring pattern?
Use coloring when basic methods and other advanced patterns like X-Wings are exhausted. Look for a candidate that forms several strong links (appears exactly twice) in many units.
What if my color chain has no contradiction?
If Simple Coloring finds no conflict, the chain remains valid but doesn't yield eliminations. You may need to try another candidate or advance to 3D Medusa to build a larger, interconnected web.