Question 107
Let $m$ and $n$ be two integers such that $m \gt n \geqslant 3$. Morgane has arranged $m$ tokens in a circle and is about to paint them using $n$ distinct colours. She wants every $n+1$ consecutive tokens to always contain at least one token of each of the $n$ colours. If she can achieve this, we say that the integer $m$ is $n$-colourable.
Prove that, for every integer $n \geqslant 3$, there are only finitely many, but at least one, integers $m$ that are not $n$-colourable, and find the largest integer $m$ that is not $n$-colourable.
Prove that, for every integer $n \geqslant 3$, there are only finitely many, but at least one, integers $m$ that are not $n$-colourable, and find the largest integer $m$ that is not $n$-colourable.