Question 17
An eight-sided polygon is drawn on a board. Mili writes an integer from $1$ to $16$ on each side and each vertex, without repetition. For each side she adds the numbers at its two endpoints and the number on the side, obtaining $8$ results. The goal is for all $8$ results to be equal. Let $S$ be the common value of the $8$ sums. Determine all possible values of $S$ and, for the smallest, give an arrangement of the $16$ numbers producing that value of $S$.