SIMON-MARAIS 2020 B4

The following problem is open in the sense that no solution is currently known to part (b). A proof of part (a) will be awarded 3 points.

Let $n \geq 2$ be an integer, and let $P_n$ be a regular polygon with $n^2 - n + 1$ vertices. We say that $n$ is taut if it is possible to choose $n$ of the vertices of $P_n$ such that the pairwise distances between the chosen vertices are all distinct.

(a) Show that if $n - 1$ is prime then $n$ is taut.

(b) Which integers $n \geq 2$ are taut?