M2. The diagram shows a circle that has been divided into six sectors of different sizes.
Two of the sectors are to be painted red, two of them are to be painted blue, and two of them are to be painted yellow. Any two sectors which share an edge are to be painted in different colours.
In how many ways can the circle be painted?
Solution
Label the sectors as in the diagram.
Suppose A is allocated a colour, which we call X. This can be done in three ways. Then neither $\mathrm{B}$ nor $\mathrm{F}$ can be coloured with $\mathrm{X}$.
- If C is coloured with $\mathrm{X}$, then $\mathrm{B}$ can be allocated a second colour in two ways. E must have the same colour as B since otherwise either D and $\mathrm{E}$ or $\mathrm{E}$ and $\mathrm{F}$ would have the same colour. Hence F and D have the third colour. This leads to 6 colourings.
- Similarly, if E is coloured with X, then F and C share a colour and B and D share the third. Again there are 6 colourings.
- If $\mathrm{D}$ is coloured with $\mathrm{X}$, then $\mathrm{E}$ and $\mathrm{F}$ have different colours and, independently, so do B and C. This leads to a further 12 colourings.

The total number of colourings is 24 .