Simple Coloring - Tracing Candidate Chains to Find Contradictions
Simple Coloring traces chains of conjugate pairs for a specific digit, coloring them in two alternating colors to discover contradictions. An advanced technique that goes beyond X-Wing for the most challenging puzzles.
The Core Concept of Coloring
Simple Coloring focuses on a single digit and exploits conjugate pairs - pairs of cells within a unit where that digit can only appear in exactly two places. These two cells have an exclusive relationship: if one is correct, the other must be wrong. By tracing these exclusive relationships as a chain and alternating between two colors, contradictions can be discovered.
The Two-Color Marking Process
Step 1: Choose a digit (e.g., 5). Step 2: Identify all units where 5 has exactly 2 candidate cells. Step 3: Color one cell of any conjugate pair as Color A, the other as Color B. Step 4: If a Color B cell forms a conjugate pair in another unit, color its partner as Color A. Step 5: Continue alternating until the chain ends. The result is two groups: Color A cells and Color B cells. Either all Color A cells are correct, or all Color B cells are correct.
Finding Contradictions
After coloring, look for two patterns. Pattern 1 (Same-color contradiction): If two cells of the same color appear in the same unit, that entire color is wrong, and the digit can be eliminated from all cells of that color. Pattern 2 (Cross elimination): If a third cell can see both a Color A cell and a Color B cell, the digit can be eliminated from that third cell. Regardless of which color is correct, that cell cannot contain the digit.
Relationship with X-Wing
X-Wing can be viewed as a special case of Coloring. An X-Wing is a rectangular pattern of 4 cells across 2 rows and 2 columns, which is equivalent to the minimal structure of two linked conjugate pairs. Coloring generalizes this concept to handle chains of arbitrary length. Puzzles that resist X-Wing may yield to longer coloring chains.
When to Apply Coloring
Coloring has high computational cost, so apply it only after exhausting all other techniques (Naked Singles, Hidden Singles, pairs, Box/Line Reduction, X-Wing). The choice of digit matters: digits with many units containing exactly 2 candidate cells form longer chains and make Coloring more effective. In Master/Extreme puzzles, this technique often provides the decisive breakthrough.
The Idea of the Conjugate Pair
At the core of coloring lies the concept of the conjugate pair. When a number's candidates within a unit exist in exactly two cells, those two cells are bound by a strong exclusive relationship: if one is correct, the other must be wrong. This relationship is derived logically from Sudoku's constraints, not from guessing. Coloring chains this certain exclusive relationship across the whole board and paints it in two colors, surfacing whether either color being correct is consistent, or whether one color must contradict. Its strength lies in making visible the logical connections between distant cells that you cannot notice by looking at a single cell.
The Two Conclusions the Two Colors Teach
Once the coloring is complete, two kinds of conclusions emerge. One is when two cells of the same color appear within the same unit. Since the same color is premised on being correct together, this means a contradiction, and that color's cells are all confirmed wrong. So you can eliminate the target number from that color's cells. The other is when there is a third cell seen simultaneously by cells of both different colors. Since one of the colors must ultimately be correct, that third cell cannot hold the target number and can be eliminated. The former is a strong conclusion negating an entire color; the latter is an elimination aimed at an individual cell - both based on certain logic.
Relation to X-Wing and Its Development
Coloring is deeply related to fish techniques, beginning with the X-Wing. The X-Wing is a pattern of four cells forming a two-row, two-column rectangle, which corresponds to the minimal structure of two conjugate pairs chained together. Coloring generalizes this idea so it can handle chains of any length. Therefore, even in complex placements the X-Wing cannot reach, coloring, which can follow long chains, may break through. Because the computational load is high, position it as a trump card after exhausting the basic techniques, and applying it by choosing a number with many units where candidates are limited to two cells makes it work efficiently.