Category: Brain Teasers

Napoleon has landed on a deserted planet with only two policemen on it. He is traveling around the planet, painting a red line as he goes. When Napoleon creates a loop with red paint, the smaller of the two encompassed areas is claimed by him. The policemen are trying to restrict the land Napoleon claims as much as possible. If they encounter him, they arrest him and take him away. Can you prove that the police have a strategy to stop Napoleon from claiming more than 25% of the planet’s surface?

We assume that Napoleon and the police are moving at the same speed, making decisions in real time, and fully aware of everyone’s locations.

First, we choose an axis, so that Napoleon and the two policemen lie on a single parallel. Then, the strategy of the two policemen is to move with the same speed as Napoleon, keeping identical latitudes as his at all times, and squeezing him along the parallel between them.

In order to claim 25% of the planet’s surface, Napoleon must travel at least 90°+90°=180° in total along the magnitudes. Therefore, during this time the policemen would travel 180° along the magnitudes each and catch him.

A thousand people stand in a circle in order from 1 to 1000. Number 1 has a sword. He kills the next person (Number 2) and gives the sword to the next living person (Number 3). All people keep doing the same until only one person remains. Which number survives?

First, we note that if the number of people is a power of 2, then the first person will survive every round. The greatest power of 2 that is less than 1000 is 512. Therefore, after 488 people die, there will be 512 remaining and the first one to kill the 489-th person will survive. This person has number 1+2×488=977.