Simple Colouring
Simple Colouring (also called single's chains) studies one digit at a time. A conjugate pair is a unit where the digit has exactly two possible cells — one of the two must be true. Starting anywhere, colour one cell of a pair blue and the other green, and keep alternating colours along every conjugate pair you can reach. The result is a network in which either every blue cell is the digit and every green is not, or the exact reverse. The two colours are the two possible worlds.
Two rules extract eliminations. Rule one: if two cells of the same colour share a unit, that colour is impossible — both cells would hold the digit and clash — so the whole colour is false, and the digit comes off every cell painted with it (equivalently, every cell of the other colour is the digit). Rule two: if an uncoloured cell sees a blue cell and a green cell, it can never hold the digit, because one of the two colours must be entirely true.
A short example of rule two: colouring the 5s, suppose r2c4 is blue, which forces r2c9 green (row pair), which forces r6c9 blue (column pair), which forces r6c2 green (row pair). The uncoloured cell r2c2 sees r2c4 (blue, same row) and r6c2 (green, same column). Whichever colour wins, one of those neighbours is a 5 — so r2c2 loses its 5.
Colouring shines when a digit has many conjugate pairs scattered across the grid — long networks make clashes and double-sightings likely. It is systematic rather than clever: mark carefully, alternate strictly, and the eliminations read themselves off.
- Only chain through conjugate pairs — units where the digit has exactly two spots. Weak links break the colour logic.
- Same colour twice in one unit kills that entire colour everywhere, not just the clashing pair.
- An uncoloured cell seeing both colours loses the digit; the coloured cells themselves stay untouched by rule two.