From Sudopedia, the free Sudoku reference guide
Supercoloring
From Sudopedia
Supercoloring is the precursor to Multi-Colors, which eventually evolved into Ultracoloring and 3D Medusa. This advanced solving technique was originally designed for computer solvers, although it can be executed by a human solver with lots of paper and patience.
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How it Works
The technique operates on all remaining candidates in the grid, excluding the givens and the solved cells. Although it considers all remaining candidates, it is still a single-digit solving technique. Interactions between different digits are not taken into consideration.
Assign Colors
Each remaining candidate receives a different color. For practical purposes, letters are used to represent these colors. In this stage, conjugate pairs are ignored. A grid with 50 remaining candidates will have 50 different colors.
Build Exclusion Matrix
Create a list of all color pairs which are mutually exclusive, because they represent the same digit in a single house. You only need half the matrix. A excludes B also implies B excludes A.
Build Conjugate Matrix
Create a list of all color pairs which form a conjugate pair in one or more houses.
Recolor Equivalent Colors
When A is conjugate to B and A is conjugate to C, B and C are equivalent. Every C is replaced by B in the results. This process is repeated until no more colors can be merged. At the end, each pair of conjugate colors represents a color cluster.
Build Implication Matrix
Using the Exclusion Matrix and the Conjugate Matrix, a third matrix is calculated. The following rule is used:
When A excludes B and B is conjugate to C, A implies C. This matrix only works in a single direction. When A implies C, it is not true that C also implies A.
Evaluate the Results
- The Contradiction rule
- When A implies B and A implies C and B excludes C, we have found a contradiction, allowing us to eliminate the candidates for color A.
- The Identity rule
- When A implies B and B implies A, A and B are equivalent and we can replace all instances of B by A. After this recoloring, we may be able to expand the Exclusion Matrix.
Supercoloring and other Coloring Techniques
Supercoloring did not survive as a solving technique, but its remnants can be found in Multi-Colors and other coloring techniques. Although these techniques use the same principles, the process is simplified so it can be performed by human solvers.
