Question 77
Let $n \geqslant 3$ be an integer. John and Mary play the following game: First John labels the sides of a regular $n$-gon with the numbers $1,2, \ldots, n$ in whatever order he wants, using each number exactly once. Then Mary divides this $n$-gon into triangles by drawing $n-3$ diagonals which do not intersect each other inside the $n$-gon. All these diagonals are labeled with number 1. Into each of the triangles the product of the numbers on its sides is written. Let $S$ be the sum of those $n-2$ products.
Determine the value of $S$ if Mary wants the number $S$ to be as small as possible and John wants $S$ to be as large as possible and if they both make the best possible choices.
Determine the value of $S$ if Mary wants the number $S$ to be as small as possible and John wants $S$ to be as large as possible and if they both make the best possible choices.