Question 169
We are given a family $\mathcal{F}$ of functions from the set $\{1, \ldots, n\}$ to itself. A sequence $\left(f_{1}, \ldots, f_{k}\right)$ of functions in $\mathcal{F}$ is said to be $\operatorname{good}$ if $f_{k} \circ f_{k-1} \circ \cdots \circ f_{1}$ is a constant function. Prove that if there exists a good sequence, there exists one with $k \leq n^{3}$.