Question 95
Let $n$ be an integer greater than or equal to $2$. Assign to each vertex of a regular $2n$-gon a distinct number chosen from $\{1, 2, \dots, 2n\}$.
(1) Show that there exists a method of assigning these numbers in such a way that the differences of the numbers assigned to every neighboring pair of vertices are all greater than or equal to $n-1$.
(2) Show that there is no way of assigning these numbers so that the differences of the numbers assigned to every neighboring pair of vertices are all greater than or equal to $n$.
(1) Show that there exists a method of assigning these numbers in such a way that the differences of the numbers assigned to every neighboring pair of vertices are all greater than or equal to $n-1$.
(2) Show that there is no way of assigning these numbers so that the differences of the numbers assigned to every neighboring pair of vertices are all greater than or equal to $n$.