Simple Colouring
Simple colouring (spelled coloring in US guides) is your first real chain technique for a single digit. It turns a run of two-cell links into a two-colour map, and from that map you can read off contradictions that no simpler technique reveals. It's also the foundation for the more powerful colouring and chain methods beyond it.
Strong links and conjugate pairs
The building block is a conjugate pair: a unit in which a digit has exactly two possible cells. One of them must be the digit, the other must not — a genuine either/or. Simple colouring picks one digit and follows these conjugate pairs across the grid, painting the two ends of each link in opposite colours, say blue and green. Because the links are strict either/ors, everything you paint blue is "on together" and everything green is "off together" — or the other way round; you don't know which yet, only that the two colours are opposites.
The two eliminations
Once a chain is coloured, two kinds of contradiction let you make eliminations:
- Colour appears twice in a unit. If two cells of the same colour share a row, column or box, that colour can't be the "on" one — two of the same digit in a unit is illegal. So every cell of that colour is false, and the digit is removed from all of them at once.
- A cell sees both colours. If an uncoloured cell can see a blue cell and a green cell for the digit, then whichever colour turns out "on", this cell is in its sights. The digit is eliminated from it.
A conjugate chain on 4 painted blue/green along strong links (solid) — the two colours are opposite states. r5c1 sees a blue 4 (r1c1, column 1) and a green 4 (r5c8, row 5), so whichever colour is "on", r5c1 loses its 4.
Step by step
- Pick a digit and mark every conjugate pair for it.
- Colour a chain of linked pairs in two alternating colours.
- Check each colour for a repeat in any unit — if found, that whole colour is false.
- Check uncoloured cells that see both colours — eliminate the digit there.
Where it leads
Simple colouring is the gateway to a whole family: X-Chains generalise it with weak links, and multi-colouring and 3D Medusa extend it further. If you've mastered the remote pair, colouring is the natural next rung on the ladder.
Common mistakes
- Colouring weak links. Simple colouring uses only strong (conjugate) links — a digit with three or more homes in a unit isn't a conjugate pair.
- Forgetting the first rule. A repeated colour in a unit kills the whole colour — an easy win that's easy to overlook.
Which Sudoku types is this best for?
The same logic applies across variants, but it pays off more in some than others.