1. On the table lie 54 piles of stones with $1,2,3, \ldots, 54$ stones. In each step, we select any pile, say with $k$ stones, and remove it from the table along with $k$ stones from each pile that has at least $k$ stones. For example, after the first step, if we select the pile with 52 stones, the piles remaining on the table will have $1,2,3, \ldots, 51,1$ and 2 stones. Suppose that after a certain number of steps, only one pile remains on the table. Justify how many stones can be in it.
1. If in each step we choose the pile with the most stones, we will gradually remove piles with $54, 53, 52, \ldots$ stones, and after 53 steps, only one pile with one stone will remain on the table.

We can prove that regardless of the procedure, the last pile will always contain a single stone. We will show that after each step, as long as there is at least one pile left on the table, the number of stones in each pile will always form the entire set $\{1, 2, \ldots, n\}$ for some natural number $n$ (we do not exclude the possibility that some numbers may correspond to multiple piles with the same number of stones). This means that there will always be at least one pile with exactly one stone on the table.

At the beginning, the number of stones in the piles forms the set $\{1, 2, \ldots, 54\}$. Suppose that after a certain number of steps, the number of stones in the piles forms the set $\{1, 2, \ldots, n\} (n \geq 2)$. If we now choose a pile with $n$ stones or a pile with one stone, the number of stones in the piles in the next step will form the set $\{1, 2, \ldots, n-1\}$. If we choose a pile with $m$ stones, where $m \notin \{1, n\}$, the number of stones in the piles in the next step will form the set $\{1, 2, \ldots, m-1\} \cup \{1, 2, \ldots, n-m\} = \{1, 2, \ldots, p\}$, where $p = \max \{m-1, n-m\}$. This proves the statement about the number of stones in the piles.

Answer: The last pile will always contain a single stone, regardless of the chosen procedure.

For a complete solution, award 6 points, of which 4 points for formulating the hypothesis that after each step, the number of stones in the piles forms the entire set $\{1, 2, \ldots, n\}$.