There's a duck in the center of a circular pond who wishes to reach the edge of the pond and thence fly away, however there's a hungry, omniscient fox at the edge of the pond who moves exactly 4 times faster than the duck and will attempt to intercept it. Can the duck escape? If so tell me precisely how. If not, prove that it's impossible.
There's a pirate ship in an alternative universe where space is discrete and one-dimensional (e.g. locations are like integers) and time is discrete (e.g. points in time are natural numbers). The pirate ship has some unknown starting location, and an unknown constant velocity. At each moment in time you can use your inter-universal cannon once to shoot some location in that universe. If you shoot the location where the pirate ship is at that time, it's destroyed! Give me an algorithm for deciding where to shoot at each time step such that you're guaranteed to destroy the ship in a finite number of steps.
There are five boxes in a row and a cat is hiding in one. Each day you get to look in one box and each night the cat must move to one of the adjacent boxes. Your job is to devise a pattern of looking in the boxes such that you can guarantee you eventually find the cat.
There are infinitely many prisoners (in the positive direction) in a line, facing the infinite end of the line, each wearing a white or black hat. They will be asked in turn what color hat they're wearing and if they say it correctly, they live. Unfortunately these prisoners are deaf so they can't hear what the prisoners behind them say, but they're not blind so they can see the infinitely many prisoners in front of them. They also are perfect logicians with unlimited memory, and can coordinate on a strategy in advance. How can you save all but finitely many of them? Hint: use the axiom of choice.
Suppose you have a complete graph of 6 points (K₆), and you color each edge either blue or red. Prove that there is either a blue triangle or a red triangle.