There are $n!$ empty baskets in a row, labelled $1, 2, . . . , n!$. Caesar first puts a stone in every basket. Caesar then puts 2 stones in every second basket. Caesar continues similarly until he has put $n$ stones into every nth basket. In other words, for each $i = 1, 2, . . . , n,$ Caesar puts $i$ stones into the baskets labelled $i, 2i, 3i, . . . , n!.$ Let $x_i$ be the number of stones in basket $i$ after all these steps. Show that $n! \cdot n^2 \leq \sum_{i=1}^{n!} x_i^2 \leq n! \cdot n^2 \cdot \sum_{i=1}^{n} \frac{1}{i} $