Multi-Colouring
Multi-colouring picks up where simple colouring stops. Simple colouring follows one chain of conjugate pairs on a single digit and colours it with two alternating colours. Multi-colouring builds two separate colour clusters for the same digit and studies how they interact — which reaches eliminations a single cluster cannot.
The idea
Each cluster is a network of strong links coloured in two shades; within a cluster, exactly one shade is true. The power comes from relating clusters. There are two classic rules. First, if a colour in cluster A can see a colour in cluster B, and also the opposite colours of the two clusters see each other, then a contradiction pins one colour false. Second — the more common one — if a colour in cluster A and a colour in cluster B both see the same outside cell, that outside cell cannot hold the digit: whichever of the two clusters is right, one of those colours is true, so the shared peer is doomed either way.
A worked example
Working the digit 8, cluster A is the conjugate pair in column 1 — r1c1 green, r5c1 yellow — and cluster B is the pair in column 8 — r1c8 blue, r5c8 red. In each cluster exactly one shade is the true 8. Now look at r1c4: along row 1 it sees the green 8 (r1c1) and the blue 8 (r1c8). Because one of green or blue must be a true 8, r1c4 can never be 8 and the candidate is removed. Neither cluster alone touched that cell; only their interaction did.
Two clusters on 8: A (green/yellow, column 1) and B (blue/red, column 8), each a conjugate pair where one shade is the true 8. r1c4 sees the green 8 and the blue 8 along row 1 — one is true, so r1c4 loses its 8.
Step by step
- Choose a digit and colour one conjugate-pair chain as in simple colouring.
- Build a second cluster from a different chain of the same digit.
- Compare the clusters for colours that see each other or a shared outside cell.
- Eliminate the digit from any cell seeing a true-bearing colour of each cluster.
Common mistakes
- Mixing clusters. Keep each cluster's two colours distinct; four colours in play, two per cluster.
- Weak links. Clusters are built from strong links only — a shared unit alone does not colour a cell.
- Forgetting the shared-peer rule. The everyday elimination is the outside cell that sees a colour from each cluster.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.