Maximum Neighbouring Sums

The vertices of a regular 10-gon are labeled V1,V2,VnV_1, V_2, \ldots V_n, which is a permutation of {1,2,,10} \{ 1, 2, \ldots, 10\}. Define a neighboring sum to be the sum of 3 consecutive vertices Vi,Vi+1 V_i, V_{i+1} and Vi+2V_{i+2} [where V11=V1,V12=V2V_{11}=V_1, V_{12}=V_2]. For each permutation σ\sigma, let NσN_\sigma denote the maximum neighboring sum. As σ\sigma ranges over all permutations, what is the minimum value of NσN_\sigma?

Details and assumptions

If the integers are written as 1,2,3,4,5,6,7,8,9,101, 2, 3, 4, 5, 6, 7, 8, 9, 10 around the circle, then the neighboring sums are 6,9,12,15,6, 9, 12, 15, 18,21,24,18, 21, 24, 27,20,1327, 20, 13, and the maximum neighboring sum is 27.

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