NL-NWO 1998 Problem 1

Consider any permutation $\sigma$ of $\{0,1,2,\dots,9\}$ and for each of the 8 triples of consecutive numbers in this permutation, consider the sum of these three numbers. Let $M(\sigma)$ be the largest of these 8 sums. (For example, for the permutation $\sigma = (4, 6, 2, 9, 0, 1, 8, 5, 7, 3)$ we get the 8 sums 12, 17, 11, 10, 9, 14, 20, 15, and $M(\sigma) = 20$ .)

(a) Find a permutation $\sigma_1$ such that $M(\sigma_1) = 13$ .

(b) Does there exist a permutation $\sigma_2$ such that $M(\sigma_2) = 12$ ?