Suppose you walk past a barber's shop one day, and see a sign that says

"Do you shave yourself? If not, come in and I'll shave you! I shave anyone who does not shave himself, and noone else."
This seems fair enough, and fairly simple, until, a little later, the following question occurs to you - does the barber shave himself?

When we finally meet the Martians, John Conway believes they are going to want to talk mathematics. He talks to Plus about his Life game, artificial life and what we will have in common with extraterrestrials.