MEMO 2011 I2

Let $n \geq 3$ 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.