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. 


Every simple closed curve that you can draw by hand will pass through the corners of some square.  The question was asked by Toeplitz in 1911, and has only been partially answered in 1989 by Stromquist.  As of now, the answer is only known to be positive, for the curves that can be drawn by hand. (i.e. the curves that are piecewise the graph of a continuous function) 

I find the result beyond my intuition. 



For details, see http://www.webpages.uidaho.edu/~markn/squares/  (the figure is also borrowed from this site)
