A lady and a monster
Here is another rather famous riddle - I've seen it several times, but only once in its full form that I quote here:
A duck is located in the center of a circular pond, near which a hungry fox is waiting. The fox cannot swim - it can only run around the pond. The duck can fly only if it gets out of the pond (don't ask, it's probably a genetically altered duck). What's the minimal ratio of the fox to duck speed such that the fox will always catch the duck? What's the duck's strategy to escape the fox if the fox is slower than that?
The lower bound of $\pi$ is immediate. Upon some further reflection one gets $\pi+1$, but I heard one can do better than that. Any ideas?