Question 182
Let $n$ be a natural number. In a row there are $n+1$ bowls, numbered from left to right with the numbers $0, 1, \ldots, n$. At the beginning there are $n$ stones in bowl $0$ and no stone in the other bowls. Sisyphus wants to move these $n$ stones into bowl $n$. To do this, in each move Sisyphus moves exactly one stone from a bowl $i$ containing $k$ stones by at most $k$ bowls to the right, i.e. into a bowl with number less than or equal to $k+i$. Let $T$ be the minimal number of moves Sisyphus needs to move all stones into bowl $n$. Show that:
$$
T \geq \left\lceil \frac{n}{1} \right\rceil + \left\lceil \frac{n}{2} \right\rceil + \ldots + \left\lceil \frac{n}{n} \right\rceil
$$
$$
T \geq \left\lceil \frac{n}{1} \right\rceil + \left\lceil \frac{n}{2} \right\rceil + \ldots + \left\lceil \frac{n}{n} \right\rceil
$$