Last Space Police question...
Sly Cooper has landed on a deserted planet with 2 policemen. He is moving around the planet, painting a line along his path, claiming any land which is surrounded by paint (the smaller area of the two resulting parts). The policemen are trying to restrict the land Sly Cooper claims as much as possible. If they encounter him, they arrest him and take him away. Can you prove that the policemen have a strategy, which prevents Sly Cooper from claiming more than 25% of the surface of the planet?
Remark: We assume that Sly Cooper and the policemen are moving with the same speed, take decisions in real time and are fully aware of everybody's locations.
P.S. Don't care much about reputation points, so decided to spread some around...
To get some institution about the problem, try to prove that one policeman alone is enough to restrict the area to 50%. Even though identical strategy may not work for the case of two policemen, it still may give some good ideas.
The solution (at least mine) does not require considering different cases. It is just a universal strategy, working no matter what the initial locations are.