I thought I'd throw a little Philosophy into the mix.. so here's one of my favorite questions. (I don't really consider this to be a paradox, but many do.)
So the story goes that the hero Theseus rode his ship in ancient Greece to slay a Minotaur who had been demanding sacrificial victims every year for generations. Years after the Minotaur was slain and Theseus was dead and gone, the people of Athens placed his ship in their city square and preserved it for many years to come.
However, wood wears and rots and decays with time. Whenever a piece of the ship became to weathered, another piece of wood was brought in to replace it. After a while, all of the wood had been replaced. Conflict then arose:
Is the ship that stands in the square still the same ship after all the wood has been replaced? If not... then at what point did it become a second ship?
Further... what then is the pile of wood that made up the original ship? And what happens if someone rebuilds a ship of that wood? Which ship is the original ship?
There is no right answer to this question. It is completely up for debate. Feel free to put forth another situation in which the ship may have become another ship (for instance, some suggest that if a thing is the sum of exactly all its parts... the ship becomes a new ship every time the tiniest chip of wood breaks off or splinters in the wind, in which case the ship might have become a new ship thousands of times before Theseus even came back from his trip.)
Extra credit points:
Think about your answer, and then how it would relate to the same problem if the ship were instead a human. Imagine that we have this dilemma in the future when human body parts are easily replaced by mechanical or bionic parts. At what point does a person cease to be the same person?
Philosophers, on your marks...