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.
Multi-coloring picks up where simple coloring leaves off. After two-colouring one strong-link chain on a digit, you find a second disjoint chain on the same digit — a separate cluster of bilocations that doesn't touch the first. Two-colour that one too, with a fresh pair of colours. Now there are four colour-classes on the grid, and the eliminations come from how the two chains interact.
The two interaction patterns
The eliminations come in two recognisable shapes.
Colour-pair sees colour-pair. If a cell of colour A1 (chain 1) sees a cell of colour B1 (chain 2), and a cell of colour A2 (chain 1) sees a cell of colour B2 (chain 2), then either both A1 and B1 are the digit, or A2 and B2 are. Whichever way, one of A1 or A2 is the digit, and one of B1 or B2 is. So any candidate of the digit that sees a cell of colour A1 and a cell of colour A2 — both colours of chain 1 — can be eliminated. Same for B1 and B2 of chain 2.
Colour-class wraps a colour-class. If every cell of colour A1 (chain 1) sees at least one cell of colour B1 (chain 2), then colour A1 cannot be the digit — because if it were, every cell B1 sees a digit-cell in its unit, which means colour B1 cannot be the digit either, but the chains are conjugate so that's a contradiction. Therefore colour A1 is eliminated everywhere it appears.
The reasoning sounds delicate. In practice multi-coloring is one of the rarer chain techniques to fire, because the conditions require two well-developed chains in the same digit's strong-link graph and at least one inter-chain link.
Why it's less common than simple coloring
Most digits don't have enough bilocations to support two disjoint chains of meaningful size. When they do, simple coloring usually fires first within whichever chain is larger. Multi-coloring is the tool of last resort when simple coloring exhausts itself but a multi-chain elimination is still available.
For a unified treatment that handles arbitrary numbers of chains and digits at once, 3D Medusa generalises both simple and multi-coloring into a single coloured graph across every digit on the grid.
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.
- 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.
- Weak link— A relationship between two cells where at most one can hold the digit. Looser than a strong link — both might be other digits — and the steady half of every chain technique.
- 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.
- 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.
Read more
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