Simple Colouring in Sudoku: A Color-Based Solving Technique
Simple Colouring is a candidate elimination technique for hard Sudoku puzzles that uses two colors to track the alternate possibilities of a single digit along a chain of strong links. When you are stuck on a puzzle with many [pencil marks](/blog/sudoku-pencil-marks) and need to find a logical breakthrough, applying two-colour reasoning to a carefully chosen candidate can reveal contradictions that force eliminations. This method is a foundational form of chaining logic that builds a visual map of a digit's potential placements, making complex inferences more manageable.
When to Use Simple Colouring
You should consider the Sudoku Colouring Technique when basic techniques like Naked and Hidden Pairs are exhausted, but the puzzle still resists. The board will be filled with pencil marks, and you need a logical step that doesn't rely on guesswork. Look for a specific candidate number (e.g., all the 5s) that appears in several cells forming a useful network of connections. The ideal setup is a candidate that forms several strong links, where it appears exactly twice in a row, column, or box. These strong links are the building blocks for creating a colour chain.
- Start with a candidate that appears frequently across the grid (6-8 times is often ideal).
- Focus on candidates that create the longest, most interconnected chain for the most powerful deductions.
How Simple Colouring Works: The Two-Color Rule
The core principle is that if two cells in the same unit (row, column, or box) are the only possible places for a digit, they form a strong link. In a true solution, exactly one of them must be correct. We assign these two cells opposite colors, say Green and Yellow, to represent these mutually exclusive 'either/or' states.
You then extend this coloring. If a Green cell is strongly linked to another cell elsewhere, that new cell must be Yellow. Continue this process, alternating colors along every strong link for your chosen candidate number. You are essentially mapping out two complete, contradictory scenarios for where that digit could possibly go. This process builds what is known as one of the fundamental Sudoku Chains.
Finding Eliminations: Type 1 and Type 2 Conflicts
The power of Simple Colouring comes from analyzing the completed color map for conflicts. There are two primary ways to achieve Sudoku Candidate Elimination.
**Type 1 (Direct Unit Conflict):** If two cells of the *same color* ever end up in the same row, column, or 3x3 box, a contradiction occurs. They cannot both be true. Therefore, the assumption that color represents a true digit is false. This means *all* cells of that color can be eliminated as candidates for that number.
**Type 2 (Elimination by Sight):** If an *uncolored* candidate cell for your digit can 'see' (is in the same row, column, or box as) two cells of *opposite colors*, it creates a logical trap. No matter which color scenario ends up being true (Green or Yellow), one of those colored cells will be the real digit, eliminating the possibility from the uncolored cell that sees both. Therefore, you can remove that candidate from the uncolored cell.
Worked Example: A Concrete Elimination
Let's trace a real pattern. Suppose we are coloring all candidate 7s. We find a strong link in row 2: only cells R2C3 and R2C7 can be a 7. We color R2C3 Green and R2C7 Yellow.
Now, cell R2C7 (Yellow) forms a strong link in its column with R5C7. Following the alternating rule, we color R5C7 Green. Cell R5C7 (Green) then forms a strong link in its box with R4C8, which we color Yellow.
Look at cell R4C3. It is an uncolored cell that contains candidate 7. Notice its position: it is in the same row as R4C8 (Yellow) and in the same column as R2C3 (Green). This is a classic Type 2 scenario. Cell R4C3 sees both a Green and a Yellow 7. Whether the final 7 is on a Green or Yellow cell, R4C3 would be in conflict with it. Therefore, we can safely eliminate candidate 7 from R4C3. This kind of chain-based elimination is a simpler precursor to more complex patterns like the X-Wing.
Practice and Next Steps
Mastering Simple Colouring trains your eye for strong links and chain logic, which is essential for advanced techniques. Practice by applying it to hard puzzles on SudokuHint.com. Use the hint engine to confirm when a colouring opportunity is present. Remember, the chain must be built exclusively on strong links (only two candidates in a unit). If a unit has three or more candidates for your number, you cannot draw a strong link through it, which may break or limit your chain.
- If your initial chain is short and yields no conflicts, try building a chain for a different candidate number.
- Always double-check that your links are truly strong (exactly two candidates in the unit) before assigning colors.
Key Facts
- ▪Simple Colouring is a Sudoku technique that uses two colors to track the alternate possibilities of a single candidate digit along chains of strong links.
- ▪A strong link exists when a candidate number appears in exactly two cells within a row, column, or 3x3 box, meaning one must be true.
- ▪The core rule is to alternate colors (e.g., Green/Yellow) with every strong link you traverse in the chain.
- ▪A Type 1 elimination occurs if two cells of the same color end up in the same unit, proving that color is false and allowing you to eliminate the candidate from all cells of that color.
- ▪A Type 2 elimination targets an uncolored cell; if it 'sees' (shares a row, column, or box with) two cells of opposite colors, the candidate can be removed from the uncolored cell.
- ▪The technique is purely logical and does not involve guessing, as it explores the direct implications of strong links.
- ▪Simple Colouring is often one of the first chaining techniques players learn before advancing to methods like X-Wing or XY-Chain.
- ▪Successful application requires a candidate digit that forms a network of strong links across the grid, often visible in harder puzzles.
Frequently Asked Questions
What is the difference between Simple Colouring and Multi-Colouring?
Simple Colouring uses one chain for a single digit. Multi-Colouring extends this by combining conclusions from multiple, separate colour chains for the same digit, leading to more complex eliminations.
Can I use any two colors?
Yes, the specific colors do not matter. The logic relies on the distinction between two opposite states, commonly labeled as Group A/Group B, On/Off, or any two distinct colors or symbols.
What if my color chain breaks?
If a candidate has three possibilities in a unit, you cannot form a strong link there. Your chain may end, or you may need to start coloring from a different strong link to build a longer, more useful chain.
Is Simple Colouring the same as an X-Wing?
No, but they are related. An X-Wing is a specific, symmetrical pattern that can often be identified and proven using Simple Colouring logic on two parallel rows or columns.