DK-GEORGMOHR 2022 Problem 5

Let $n > 2$ be an integer. The numbers $1, 2, . . . , n$ are written at the vertices of an $n$ -gon in that order. A move consists of choosing two adjacent vertices and adding $1$ to the numbers written there. Determine all n for which it is possible to achieve that all numbers are identical after a finite number of moves.