AUSTRIAN-POLISH 1994 Problem 4

The vertices of a regular $n + 1$ -gon are denoted by $P_0,P_1,...,P_n$ in some order ( $n \ge 2$ ). Each side of the polygon is assigned a natural number as follows: if the endpoints of the side are $P_i$ and $P_j$ , then the assigned number equals $|i - j |$ . Let S be the sum of all $n + 1$ assigned numbers. (a) Given $n$ , what is the smallest possible value of $S$ ? (b) If $P_0$ is fixed, how many different assignments are there for which $S$ attains the smallest value?