Here is the problem:
For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $\dfrac{\binom{n}{2}}{3}$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle?
Trivially $n\equiv 0 \text{ or } 1 \pmod{3}$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks