# 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.