A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language The following is a quote from Surely you're joking, Mr. Feynman.  The question is: are there any interesting theorems that you think would be a good example to tell Richard Feynman, as an answer to his challenge?  Theorems should be totally counter-intuitive, and be easily translatable to everyday language.  (Apparently the Banach-Tarski paradox was not a good example.)

Then I got an idea.  I challenged
  them: "I bet there isn't a single
  theorem that you can tell me - what
  the assumptions are and what the
  theorem is in terms I can understand -
  where I can't tell you right away
  whether it's true or false."
It often went like this: They would
  explain to me, "You've got an orange,
  OK? Now you cut the orange into a
  finite number of pieces, put it back
  together, and it's as big as the sun.
  True or false?"
"No holes."
"Impossible!
"Ha! Everybody gather around! It's
  So-and-so's theorem of immeasurable
  measure!"
Just when they think they've got
  me, I remind them, "But you said an
  orange! You can't cut the orange peel
  any thinner than the atoms."
"But we have the condition of
  continuity: We can keep on cutting!"
"No, you said an orange, so I
  assumed that you meant a real orange."
So I always won. If I guessed it
  right, great. If I guessed it wrong,
  there was always something I could
  find in their simplification that they
  left out. 


The Fold-and-Cut Theorem is pretty unintuitive. http://erikdemaine.org/foldcut/
