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.
Murderous assassins are deploying venomous spiders against our beloved King where he sleeps in his bedchambers. These spiders are somewhat unusual: For one, they are “point spiders”: they have no volume, surface area, or dimensionality. They can also traverse any continuous surface, and can fall straight down from one surface to another. Crucially, they are allergic to water and will perish immediately upon contact. Your job is to protect the King by remodeling his bedchambers. Your design must be physically constructible and static. It may use fully supported still water, but no moving parts. The King must be able to enter and leave without getting wet (though he's willing to step over water to make it into his bed). Most importantly, it must be impossible for spiders to reach him while he's in his bed. You should assume the spiders will approach your construction both by crawling along the ground to reach it and by falling out of the sky. Your construction must not be dependent on infinitesimal precision – e.g. the King must remain safe even if a gust of air blows a falling spider epsilon distance to the side. Draw up a blueprint and prove that it is safe.