# Maximum Neighbouring Sums

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 $$N_\sigma$$?

Details and assumptions

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

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