3D Medusa
An advanced colouring technique that two-colours every digit's strong-link graph at once, finding cross-digit eliminations that single-coloured chains miss.
3D Medusa is the colouring technique that generalises simple coloring and multi-coloring into a single unified scheme. Where simple coloring two-colours one digit's strong-link graph and multi-coloring extends to multiple disjoint chains on the same digit, 3D Medusa colours every digit's graph simultaneously — and adds the cross-digit constraint that within any cell, each candidate inherits a colour and the colours must be consistent.
What "3D" means here
The "3D" in the name refers to the conceptual third axis. Simple coloring runs in two dimensions: the rows and columns of the grid, two-coloured for one digit. Multi-coloring runs in two dimensions but on multiple chains. 3D Medusa adds a third axis — the digit dimension — and propagates colour information across digits whenever a cell has multiple candidates.
The rule is straightforward in concept. If a cell has two candidates a and b, and a is coloured A in its strong-link graph, then b inherits the opposite colour B (because exactly one of the cell's candidates is the answer, and the strong-link colour A means "this candidate is the digit if every A-coloured cell is"). The colour propagation continues across the grid, picking up implications between digits via shared cells.
The eliminations
Three families of elimination fire from a 3D Medusa:
- Same-colour collision in a unit. Two cells of the same colour share a unit — the colour is impossible, eliminate it everywhere.
- Two opposite colours in one cell on different digits. A cell holds candidates a (colour A1) and b (colour B2). A1 and B2 are constrained: if A1 is the truth, the cell must be a; if B2 is the truth, the cell must be b. Other candidates of the cell can be eliminated.
- A candidate seeing both colours of its own digit. Standard simple-coloring elimination, generalised across the unified colouring.
The first two extend single-digit colouring's reach significantly. A 3D Medusa often produces several eliminations in one pattern, where simple coloring would have produced only one or two on each digit's chain.
When it earns the cost
3D Medusa is heavy to set up by hand. Two-colouring every digit's strong-link graph and tracking cross-digit colour inheritance is a lot of bookkeeping. Most solvers reach for it only when AIC and the named single-digit patterns have been exhausted. Software solvers run it cheaply because the colour propagation is mechanical; that's where 3D Medusa earns its place in the technique hierarchy of the more thorough open-source solvers.
For most extreme puzzles, the eliminations produced by a 3D Medusa overlap with those a careful AIC search would have found anyway. The technique's value is having a single unified algorithm that finds them without enumerating chains by hand.
See also
- Simple coloring— A technique that two-colours the strong-link graph of a single digit, then eliminates candidates that see both colours — the entry point into chain reasoning.
- Multi-coloring— An extension of simple coloring to two or more disjoint chains on the same digit, finding eliminations that fire when the chains interact at a distance.
- Alternating Inference Chain (AIC)— The general-purpose chain technique. Alternates strong and weak links along a sequence of candidates, eliminating a digit from any cell that sees both endpoints' candidates.
- Strong link— A relationship between two cells in a unit where a digit must occupy exactly one of them — the basic primitive that hidden singles, X-wings, and chain reasoning all rest on.
Read more
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