You permute each of the digits $0, 1,2, \dots, 9$ and write them in a single row.
A) Show that no matter what the permutation, some 3 adjacent elements of the row sum to at least $13$.
B) Can you find a permutation where no 3 adjacent elements sum to more than $13$?
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A) Say the permutation is $x_0, x_1, \dots, x_9$.
Now one of $x_0$ or $x_9$ is $< 9$. We can assume $x_0 < 9$.
Let $S_i = x_i + x_{i+1} + x_{i+2}$.
Since $x_0 < 9$ we must have that $S_1 + S_4 + S_7 > 36$ and thus $\text{max} \{S_1, S_4, S_7\} > 12$.
B) 9 3 1 7 4 2 6 0 5 8
