Tier 3 · first asked for by Tough
Simple Colouring
Follow one digit through its two-cell units, colour the chain alternately, and one colour must be true.
Also called Singles Chains, Colouring.
Pick a digit and find every unit where it has exactly two possible cells. Those pairs are strong links: one of the two is certainly the digit. Chain the links together and colour the cells alternately, so that all cells of one colour are true and all of the other false — you do not yet know which way round.
Two conclusions follow. If two cells of the same colour share a unit, that colour would place the digit twice, so that colour is false and every cell in it loses the digit. And any cell outside the chain that can see both colours loses the digit too, because one of the colours is true whichever way it falls.
A worked example
A real position, with the pencil marks a solver would have at that point.
the pattern · what it rules out · 5 a candidate that goes
What it proves. Colouring on 7: one colour of the chain would place 7 twice in a unit, so 7 leaves R1C9, R3C8.