There are already $N$ people sitting around a circular table with 60 chairs. What is the smallest possible value for $N$ if the next person to sit down will have to sit next to someone?
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. (In the figure, occupied chairs are represented by black squares and unoccupied chairs by white squares.)

![](https://cdn.mathpix.com/cropped/2024_05_01_280b19a87fa67143351eg-072.jpg?height=309&width=292&top_left_y=1950&top_left_x=1590)

We can then think of the chairs in ordered groups of 3 chairs, where the third is already occupied. Therefore, the smallest value of $N$ is $60 \div 3=20$.