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.

8888✗8

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

  1. Choose a digit and colour one conjugate-pair chain as in simple colouring.
  2. Build a second cluster from a different chain of the same digit.
  3. Compare the clusters for colours that see each other or a shared outside cell.
  4. 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.

Great fit Classic Sudoku Extends simple colouring to catch single-digit eliminations across two clusters. → Limited Killer Sudoku Rarely reached before cage arithmetic cracks Killer. → Works Sudoku X Diagonal conjugate pairs create extra colouring links. → Limited Samurai Sudoku Cluster bookkeeping across five grids is unwieldy. →

Where to practise this

This technique starts paying off at Expert, which is where puzzles first stop yielding to anything simpler. Reading about a pattern and spotting it on a live grid are different skills - the second one only comes from playing.

Play Expert Sudoku About Expert Sudoku

Related techniques

Simple Colouring - Sudoku Chain Technique Colour a digit's conjugate-pair chain in two alternating colours; a colour that appears twice in a unit is false, and cells seeing both colours lose the digit. X-Wing Sudoku Technique Explained One digit confined to the same two columns across two rows (or vice versa) - eliminate it from the rest of those lines. XY-Wing Sudoku Technique (Y-Wing) Explained A pivot cell with two candidates links two pincer cells; the digit they share is removed from any cell both pincers see. Swordfish Sudoku Technique Explained One digit confined to the same three columns across three rows (or vice versa) - eliminate it from the rest of those columns.
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