Suppose you have two bottles with equal perimeter, thickness and height. One bottle is cilindrically shaped, the other one has the shape of a rectangular box with a square shaped bottom. In which of the bottles fits more milk?

What is the mean distance between two randomly choosen points within a sphere of radius 1?

On the yearly fair you can play as often as you like the game "rat race". In this game a rat is freed at the beginning of a circuit. The rat will then run to hole A, B or C, every hole with equal chance. Before the rat is freed, you can bet any amount you like on the hole the rat will run into. If you choose the right hole, you'll get back twice the amount you've put a stake. If you're wrong you loose your money. Suppose you carry an infinite amount of money. Will you in that case always loose money playing this game, or does there exist a strategy to win money. Does this strategy also work in the more realistic situation when you carry a limited amount of money or if there is a maximum limit on the bet?