Multi-Colouring
Short answer: build a colouring chain for one shape, then build a second one for the same shape somewhere else. If a cell of one label in the first chain sees a cell of one label in the second, those two labels cannot both be true — so at least one of their opposites is, and any cell seeing both opposites loses the shape.
What multi-colouring looks like
Simple colouring works inside one chain and stops when the chain runs out. Most grids have more than one chain for a shape, though, and they are not independent: a single sighting between them ties the two together and lets each one constrain the other. That is all multi-colouring is — no new kind of reasoning, just refusing to treat the chains separately.
Follow the logic slowly, because it is easy to state and easy to get backwards. In any chain, exactly one of its two labels is entirely true. Label A of chain one sees label C of chain two, so A and C cannot both be true. Therefore B or D is true. A cell that sees a B cell and a D cell is dead either way.
How to use multi-colouring
- Build the chains. Pick one shape and mark every unit where it has exactly two possible cells. Link those pairs and label them alternately. You will usually end up with two or three separate chains rather than one.
- Look for a sighting between chains. Find a cell in one chain that sees a cell in another. Note which label each of them carries — those two are the labels that cannot both be true.
- Take the opposites. The elimination is driven by the other two labels. At least one of them is entirely true, though you never learn which.
- Strike what sees both opposites. Remove the shape from every cell outside both chains that sees at least one cell of each opposite label. Cells inside the chains are settled by the chains themselves.
Two related rules are worth knowing even though they rarely produce a move on their own: if two cells of the same label share a unit, that label is false outright and its opposite is true everywhere; and if one label sees both labels of another chain, that label is false. Both are checks worth making while you are already looking at the chains.
Multi-colouring vs the fish
When nothing else is left, the choice is usually between this and a finned Jellyfish — and multi-colouring is normally the better bet. Building chains is mechanical, and a mistake tends to announce itself as a contradiction rather than as a silently wrong elimination, whereas a four-line fish asks you to enumerate sixteen cells correctly and gives no warning if you do not. The fish is the right call only when the shape has too few conjugate pairs for the chains to reach anywhere.
Find Multi-Colouring in your own grid
Paste a puzzle and this page will look for Multi-Colouring in it — the same pattern shown above, in your grid instead of ours. Nothing is uploaded; it runs in your browser.
Frequently asked questions
How is this different from simple colouring?
Simple colouring uses one chain and strikes cells that see both of its labels. Multi-colouring uses two chains and the single link between them, which reaches cells that neither chain could touch alone.
Why can I not just merge the two chains?
Because the link between them is not a conjugate pair. Chain links mean "exactly one of these two is the shape"; a sighting only means "not both". That is a weaker relationship, and merging on it would make the labels claim more than they have earned.
How many chains should I build?
Build all of them for the shape you are working on — it costs nothing once you have found the conjugate pairs, and the useful link is often between the two smallest chains rather than the long one you were concentrating on.
Is this the same as 3D Medusa?
No. Medusa colours several shapes at once and links cells through bivalue cells as well as conjugate pairs. Multi-colouring stays on one shape throughout, which makes it far easier to do by hand and far harder to get wrong.
Is multi-colouring deprecated?
Upstream, yes. SudokuWiki dropped it from its solver, on the grounds that in every case it tested a simpler strategy could reach the same result. That solver also carries X-Cycles, Nice Loops and 3D Medusa, none of which form9 implements — so the bypass it relies on is not available here. On the position above, the elimination this page teaches is one that none of the twenty easier techniques form9 knows can reach. It is fair to call it optional: removing it from our solver changes no outcome. It is not fair to call it useless.