21. Each of the integers $1,2,3, \ldots, 9$ is assigned to each vertex of a regular 9 -sided polygon (that is, every vertex receives exactly one integer from $\{1,2, \ldots, 9\}$, and two vertices receive different integers) so that the sum of the integers assigned to any three consecutive vertices does not exceed some positive integer $n$. What is the least possible value of $n$ for which this assignment can be done?
Problem 21. There is an assignment of the integers $1,2,3, \ldots, 9$ to the vertices of the regular nonagon that gives $n=16$. Let $S$ be the sum of all sums of the integers assigned to three consecutive vertices. If there are integers assigned to three consecutive vertices whose sum is at most 14, then $S<135$, which is a contradiction. Thus, every sum of the integers assigned to three consecutive vertices is equal to 15 . Consider $a, b, c, d$. Then $a+b+c=15$ and $b+c+d=15$, which implies that $a=d$, a contradiction again.