A room has n empty chairs in a row. You choose a chair and sit it in it. Now someone else comes in and sits as far away as possible from everyone in the room or an arbitrary choice of furthest chair if there is more than one. This happens repeatedly with the rule that no two people can ever sit next to each other. It finishes when there is no more space to sit.
Your goal of course is to choose your chair initially so that as many people as possible can sit in the room.
Say n=2. Wherever we sit we end up with exactly one person sat down.
For n=3. If we sit in the first chair, we get full, empty, full in the end. But if we set the chair bit we get empty, full, empty which is not optimal.
For n=4. Wherever we sit we end up with two chairs with people on them.
For n=5. Sitting in the first chair gives us full, empty, full, empty,full with three people sat down in the end.
For n=7. We need to sit in the third chair it seems.
What is the simplest rule for choosing where to sit which will always give an optimal answer?