2014-09-30

Maximum Enjoyment of the Cinema

Bengt has gone to see a film at the small local cinema. This particular cinema doesn't have seat reservations, so you go inside and pick a seat that's free. Bengt isn't a very sociable person, so he doesn't want to sit next to anyone. If the row is empty, he sits on one end, and if anyone sits in the row already, he sits as far from them as possible. If he ends up sitting next to someone, because there are no empty seats that aren't next to anyone, he will be grumpy and not enjoy the film.
  1. Imagine everyone thinks like Bengt. What numbers of seats per row would be the most suitable? We want to avoid people being grumpy, but we also want the density of people to be as high as possible, so as not to waste space. Ideally, thus, we want to make it so that for some number of guests, every second seat will be occupied. This is true for certain numbers of seats - three, for example, but not four. Which are these numbers?
  2. What if the seats would be in a circle, like... some sort of circus, or something?
  3. If you're up for it, try to extend the problem to two dimensions. Or more, why not.