Question 16
Old Zhao is so bored during his military service that he draws $n + 1$ cells in a row on the ground ($n$ is a positive integer), numbered from left to right as cell 0 to cell $n$. Initially, cell 0 contains $n$ stones and all other cells are empty. In each turn, Old Zhao first chooses a non-empty cell; suppose it contains $k$ stones. Then he takes one stone from the chosen cell and moves it to the right by at most $k$ cells (the stone may not go beyond the rightmost cell). Old Zhao's goal is to put all the stones into cell $n$.
Prove that the number of turns Old Zhao needs to achieve his goal is at least
$$
\left\lceil \frac{n}{1} \right\rceil + \left\lceil \frac{n}{2} \right\rceil + \left\lceil \frac{n}{3} \right\rceil + \cdots + \left\lceil \frac{n}{n} \right\rceil,
$$
where $\lceil x \rceil$ denotes the smallest integer not less than $x$.
Prove that the number of turns Old Zhao needs to achieve his goal is at least
$$
\left\lceil \frac{n}{1} \right\rceil + \left\lceil \frac{n}{2} \right\rceil + \left\lceil \frac{n}{3} \right\rceil + \cdots + \left\lceil \frac{n}{n} \right\rceil,
$$
where $\lceil x \rceil$ denotes the smallest integer not less than $x$.