Multi Coloring extends Simple Coloring to a situation where the same digit forms more than one disconnected conjugate-pair network.
Simple Coloring gives one component two opposite colors. Multi Coloring keeps separate color pairs for separate components, then studies how those components interact through ordinary Sudoku visibility.
The key is not “use more colors because the pattern is harder.” The key is:
two independently valid either/or networks can constrain one another.
Prerequisite: understand Simple Coloring first
For one digit X, a house containing exactly two X candidates creates a conjugate pair — a Strong Link.
Color those candidates opposite colors and propagate through connected Strong Links. In one connected component, one color is globally true and the other globally false.
Simple Coloring then uses contradictions such as Color Wrap or off-chain candidates that see both colors.
Multi Coloring begins when another disconnected component for the same digit exists.
Why disconnected components need separate color pairs
Suppose component A uses colors A1/A2 and component B uses B1/B2.
You cannot assume A1 corresponds to B1.
There is no logical connection yet between the two components. Each has its own polarity:
- either A1 is true and A2 false, or vice versa;
- either B1 is true and B2 false, or vice versa.
A cross-component relationship is required before any colors can constrain one another.
The first Multi Coloring interaction
If a cell colored A1 sees a cell colored B1, those two candidates cannot both be true.
That weak relationship has a useful consequence:
- if A1 is false, A2 must be true;
- if B1 is false, B2 must be true;
- because A1 and B1 cannot both be true, at least one of their opposites A2/B2 must be true.
A candidate outside the color networks that sees all relevant A2 and B2 possibilities can therefore be eliminated.
This is the characteristic Multi Coloring move: use one cross-component weak link to create a new either/or statement across opposite colors.
A second interaction: one color sees both colors of another component
Suppose a candidate in component A with color A1 sees:
- at least one B1 candidate;
- at least one B2 candidate.
Component B must choose either B1 or B2 as true. Therefore A1 would see a true X in either case.
A1 cannot be true.
So the entire A1 color can be eliminated, and A2 becomes true throughout component A.
This resembles a Color Wrap, but the contradiction is supplied by another disconnected component rather than by two same-colored candidates inside one house.
Worked Multi Coloring example
Multi Coloring vs Simple Coloring
Simple Coloring operates inside one connected component.
Multi Coloring adds interactions between disconnected components.
Do not use four colors in one network just because it looks complex. If the Strong Links connect, one two-color polarity is enough.
Multi Coloring vs X-Chains
Both use Strong/Weak inference for a single digit.
Multi Coloring is a network visualization approach. X-Chains are a path approach.
A deduction may be expressible both ways. Use whichever representation makes the proof easiest to verify.
Multi Coloring vs 3D Medusa
Multi Coloring stays on one digit. It connects separate components for that digit.
3D Medusa crosses between different digits through bivalue cells as well as conjugate pairs, creating one multi-digit two-color network.
That makes 3D Medusa substantially more general.
Common mistakes
Giving disconnected components the same polarity
Colors are local until a logical link relates the components.
Using ordinary peers as if they were Strong Links
Two candidates seeing each other create a Weak relationship, not a conjugate pair unless the house has exactly two candidates for the digit.
Eliminating from a target that sees only one forced branch
A target must be contradicted by every remaining color alternative used in the proof.
Mixing different digits
Traditional Multi Coloring is a single-digit technique. Multi-digit coloring belongs to 3D Medusa/AIC-style reasoning.
Practical search strategy
- choose a digit with several conjugate pairs;
- build the first connected color component;
- if conjugate pairs remain disconnected, build another component with a new color pair;
- inspect visibility between colored candidates across components;
- test the standard cross-component contradiction patterns;
- after any elimination, rebuild the candidate state rather than preserving stale colors.
FAQ
What is Multi Coloring in Sudoku?
It is a single-digit technique that colors multiple disconnected conjugate-pair components separately and then uses relationships between those components to make eliminations.
Why not use the same two colors everywhere?
Disconnected components have independent polarity. Without a link, there is no reason the first color in one component corresponds to the first color in another.
Is Multi Coloring harder than Simple Coloring?
Usually, because you must track multiple independent either/or networks and their interactions.
Is Multi Coloring still useful if I know Chains?
Yes. Coloring can make a single-digit network visually obvious even when the equivalent Chain notation would be longer.
What to learn next
Continue to 3D Medusa if you want coloring to cross between different digits, or to X-Cycles and AIC if you prefer a formal chain/loop language.