3D Medusa

3D Medusa is a two-color Sudoku network that extends coloring beyond a single digit.

Simple Coloring follows conjugate pairs for one candidate value. 3D Medusa can also move through bivalue cells, so the network may contain many different digits while preserving one global two-state polarity.

If one color is true, every candidate of that color in the connected network is treated as true and every opposite-colored candidate false. If the opposite color is true, the states reverse.

The technique succeeds when one polarity creates a contradiction, or when an uncolored candidate is impossible under both polarities.

The two link types that build a Medusa

3D Medusa grows through Strong relationships of two forms:

Conjugate pair in a house

For one digit, if a row, column, or box has exactly two possible locations, one must be true if the other is false.

Color them opposites.

Bivalue cell

If one cell has exactly two candidates {X,Y}, one must be true and the other false.

Color those two candidates opposites too.

By alternating through houses and cells, the color network can change digits — the “third dimension” missing from single-digit coloring.

Why the whole connected network shares one polarity

Every color edge represents an either/or relationship.

Once you choose one colored candidate as true, every Strong relationship propagates a forced opposite state. Because all nodes are connected, that initial choice determines the state of the entire component.

You do not know which polarity is the real one, but you know there are only two consistent possibilities.

That is enough to test contradictions.

Core contradiction: same color twice in one cell

If two candidates with the same color appear in one cell, that color cannot be the true polarity. A Sudoku cell cannot contain two digits.

Eliminate every candidate of that color in the network and confirm the opposite color.

Core contradiction: same digit/same color twice in one house

If two candidates for the same digit with the same color see each other in a row, column, or box, they cannot both be true.

Again, that color is false throughout the network.

This is the multi-digit equivalent of a Color Wrap.

Off-network elimination: a cell contains both colors

Suppose an uncolored candidate Z shares its cell with one colored candidate of color A and can see an opposite-colored Z somewhere else in the grid.

• if A is the true polarity, the other candidate in the cell blocks Z;
• if B is the true polarity, the visible colored Z blocks it.

Z is impossible in either case and can be eliminated.

3D Medusa has several named rule presentations, but they all reduce to the same principle:

test the target against both possible global color states.

Emptying a cell or house under one polarity

Another contradiction occurs if choosing one color false would remove every candidate from a cell, or every location for a digit in a house.

A valid Sudoku cannot leave:

• an unsolved cell with no candidate;
• a row/column/box with no possible place for a required digit.

Therefore the polarity causing that state is impossible.

Worked 3D Medusa

3D Medusa vs Simple Coloring

Simple Coloring:

• one digit;
• Strong Links through houses;
• usually Color Trap / Wrap style deductions.

3D Medusa:

• multiple digits;
• Strong Links through houses and bivalue cells;
• more kinds of contradiction and off-network elimination.

3D Medusa is therefore not “Simple Coloring with more paint.” It has a larger inference graph.

3D Medusa vs AIC

The same logic can often be represented as an Alternating Inference Chain or Nice Loop.

AIC emphasizes a path of alternating Strong and Weak inferences. Medusa emphasizes the entire two-state connected network.

If the grid contains many bivalue/conjugate links, the network view may reveal several eliminations at once.

3D Medusa vs Multi Coloring

Multi Coloring stays on one digit but manages disconnected components.

3D Medusa changes digits through bivalue cells inside a connected two-color component.

The visual similarity should not hide that structural difference.

Common mistakes

Coloring through a non-Strong relationship

A normal peer relationship is Weak. You cannot propagate opposite colors through it unless the relevant state is actually conjugate/bivalue.

Combining disconnected components

One Medusa component has one polarity. A disconnected component has an independent polarity unless another inference connects them.

Assuming colored means true

Colors are labels, not truth values, until a contradiction decides the polarity.

Forgetting candidate-level coloring

Two different candidates in the same cell can have different colors. Medusa colors candidates, not just cells.

Keeping stale coloring after eliminations

The graph may change when candidates disappear. Revalidate the Strong Links.

FAQ

What does 3D mean in 3D Medusa?

It describes extending single-digit coloring across different candidate values through bivalue cells, conceptually adding a digit dimension to row/column/box coloring.

Is 3D Medusa guessing?

No. Both polarities are logical alternatives of the same Strong-Link network; eliminations are made only when one polarity contradicts Sudoku or when a target fails under both.

Do I need Multi Coloring first?

Not strictly, but Simple Coloring and Strong/Weak Links are essential prerequisites.

Is 3D Medusa the same as AIC?

No, but many Medusa deductions can be expressed as AIC/Nice Loop logic. They are different representations of closely related inference structure.

What to learn next

Move to X-Cycles for single-digit loops, Grouped AIC for more general grouped inference, or ALS if you want nodes larger than individual candidates.