Supercoloring
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.
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.
