Simple Colouring - Sudoku69

Simple Colouring takes one digit, chains together every place where that digit is squeezed into two cells, and paints the chain in two alternating colours. One colour is true and the other is false. You never find out which is which, and you do not need to.

What the pattern is

A conjugate pair is a unit — one row, column or box — in which a digit has exactly two remaining candidate cells. One of the two must hold the digit, so the pair works like a switch: turn one off and the other turns on.

Conjugate pairs often share a cell. When they do they form a chain. Colour the first cell blue, colour its partner green, then keep alternating at every link. Every blue cell now carries the same truth value, and every green cell carries the opposite one.

When to look for it

Pencil marks are complete and singles, pairs and X-Wings have run dry.
A digit is already placed in five or six boxes, so its leftover candidates are sparse.
Two conjugate pairs on the same digit share a cell. That is a chain of three cells, and three is enough.

How to apply it, step by step

Pick one digit and mark every unit where it has exactly two candidates.
Start at any cell of one such pair and give it colour A.
Give its partner colour B, then follow every other pair running through those cells, alternating as you go.
Stop when the chain stops growing. Candidates of that digit that never joined stay uncoloured.
Test the finished colouring against the two rules below.

Rule	What you see	What it gives you
Colour clash	Two cells of the same colour inside one unit	That colour is false, so the other colour is the answer in every cell it covers
Colour trap	An uncoloured candidate that sees a cell of each colour	That candidate is impossible and comes off

The clash rule is the stronger of the two because it places digits outright. The trap rule only removes candidates, but it fires far more often.

A worked example

Only the digit 7 matters here, and only the top three rows.

Example

        c1 c2 c3   c4 c5 c6   c7 c8 c9
r1       7  .  .    .  .  .    7  .  .
r2       .  .  .    7  7  .    7  7  7
r3       .  .  7    .  .  .    .  .  7

7 = still possible    . = ruled out

Three conjugate pairs sit in that band: row 1 (c1 and c7), row 3 (c3 and c9), and box 1 (r1c1 and r3c3).

Colour r1c1 blue. Box 1 makes r3c3 green. Row 1 makes r1c7 green. Row 3 makes r3c9 blue. Blue is now r1c1 and r3c9; green is r3c3 and r1c7.

Box 3 contains one cell of each colour, r1c7 and r3c9. The other three 7s in that box — r2c7, r2c8 and r2c9 — all see both colours, so 7 comes off each of them. Whatever happens, row 2 must take its 7 in c4 or c5.

Common mistakes

Linking two cells that merely share a unit. A link needs a conjugate pair and nothing weaker.
Forgetting box links. Most useful chains bend through a box rather than running along rows and columns only.
Trapping a cell that sees the same colour twice. Two blues prove nothing; you need one of each colour.
Treating a clash as the end of the chain. It kills one colour and confirms the other, which is usually several placements at once.
Reusing colours across digits. Every digit gets its own independent colouring.

Where you meet it

Simple Colouring is the first single-digit chaining technique, and it starts to matter at hard classic Sudoku, where wings and fish no longer finish the grid on their own. Once your chains need links that are weaker than a conjugate pair, the next step up is X-Cycles.