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Permutation of 0, 1, 2, ..., 9

Source: Dutch Mathematical Olympiad 1998

October 29, 2005

Problem Statement

Consider any permutation σ\sigma of {0,1,2,…,9}\{0,1,2,\dots,9\} and for each of the 8 triples of consecutive numbers in this permutation, consider the sum of these three numbers. Let M(σ)M(\sigma) be the largest of these 8 sums. (For example, for the permutation σ=(4,6,2,9,0,1,8,5,7,3)\sigma = (4, 6, 2, 9, 0, 1, 8, 5, 7, 3) we get the 8 sums 12, 17, 11, 10, 9, 14, 20, 15, and M(σ)=20M(\sigma) = 20.) (a) Find a permutation σ1\sigma_1 such that M(σ1)=13M(\sigma_1) = 13. (b) Does there exist a permutation σ2\sigma_2 such that M(σ2)=12M(\sigma_2) = 12?
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