A circular table has 60 chairs around it. There are $N$ people sitting in these chairs in such a way that the next person to sit down will have to sit next to someone. What is the smallest possible value for $N$?

## Solutions from List 5
If the next person to sit down will have to sit next to an occupied chair, this means there are at most 2 consecutive unoccupied chairs. See the figure: the occupied chairs are represented by white squares and the unoccupied ones by black squares. We can then think of the chairs

![](https://cdn.mathpix.com/cropped/2024_05_01_7edf59f8d27fef74e9deg-6.jpg?height=474&width=439&top_left_y=1479&top_left_x=1391)
in groups of 3 with the third one being occupied. Therefore, the smallest value of \( N \) is \( 60 \div 3 = 20 \).