Problem 11.2. Along a road, 25 poles are standing in a row. Sometimes a goldfinch lands on one of the poles, and immediately a goldfinch flies away from one of the adjacent poles (if there was at least one goldfinch on the adjacent poles at that moment). Also, no more than one goldfinch can sit on each pole.

Initially, there are no birds on the poles. What is the maximum number of goldfinches that can be on the poles at the same time?
Answer: 24.

Solution. First, we will show that it is impossible to occupy all the poles. Suppose this happened. Consider the last tit that sat down. Since it occupied the last unoccupied pole, there must have been an occupied pole next to it. Consequently, the tit sitting on that pole must have flown away. Contradiction.

Now, let's provide an example of how the tits could occupy 24 poles. For convenience, let's number all the poles in order. Let the first tit sit on the first pole. Suppose all poles from 1 to $k$ are occupied, where $k \leqslant 23$. We will show how to achieve a situation where all poles from 1 to ( $k+1$ ) are occupied. By sequentially changing $k$ from 1 to 23, we will obtain a situation where the first 24 poles are occupied.

Suppose all poles from 1 to $k$ are occupied, then the next tit sits on the $(k+2)$-th pole (where $k+2 \leqslant 25$), and the next tit after that sits on the $(k+1)$-th pole, and the tit from the $(k+2)$-th pole flies away. We obtain a situation where all poles from 1 to ( $k+1$ ) are occupied.