MathDB

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.

Problem Statement