MathNet 2019-part-02 — 184 problems (EN–VI) Question 139

A positive integer $n$ is good if each side and diagonal of a regular $n$-gon can be coloured in some colour so that for each pair of vertices $A$ and $B$ there is exactly one vertex $C$, different from $A$ and $B$, such that the segments $\overline{AB}$, $\overline{BC}$ and $\overline{CA}$ have the same colour.
Which of the numbers 7, 8, 9, 10, 11 and 12 are good, and which are not?
(Ilko Brnetić)