Colouring

The Colouring technique is based on the idea of conjugates; If there are only two cells in a unit with a candidate digit, one cell must contain the digit and the other must not. Find a conjugate pair and colour one cell red, and the other green. Then find any conjugates for those two cells, and alternate the colour of each conjugate. Repeat until there are no conjugates left. The candidate digit must be in either all the red cells or all the green cells. If it would be impossible for the digit to be in one colour, it must be in the other.

Consider the sudoku below.

3 5 8 7 4 2 9
8 7 2 3 9 4 6 5 1
5 9 4 6 2 1 8 3 7
 2
1 9 4 3 7
 2
4
 2
7 1 3
 2
9 6
3 5 9 2 7 6 1 4 8
9 3 6 2 1
 2
6 5 4 1 9
 2
8 3
 2
 2
5 3 9 6
 2

The cells r4c1 and r4c9 are conjugates, so we can start the Colour Map.
The cells r4c1 and r5c2 are conjugates, so we can update the Colour Map.
The cells r5c2 and r5c7 are conjugates, so we can update the Colour Map.
The cells r5c2 and r9c2 are conjugates, so we can update the Colour Map.
The cells r5c7 and r8c7 are conjugates, so we can update the Colour Map.
The cells r8c7 and r8c1 are conjugates, so we can update the Colour Map.

At this point, there are two red cells in the bottom-left block so the digit 2 cannot exist in the red cells. Therefore, the 2 must exist in the green cells.