Question

The vertices of a regular 10-gon are labeled V_1, V_2, \ldots V_n, which is a permutation of \{ 1, 2, \ldots, 10\}. Define a neighboring sum to be the sum of 3 consecutive vertices V_i, V_{i+1} and V_{i+2} [where V_{11}=V_1, V_{12}=V_2]. For each permutation \sigma, let N_\sigma denote the maximum neighboring sum. As \sigma ranges over all permutations, what is the minimum value of N_\sigma?