Reading the autobiography of Richard Feynman, I struck upon the following paragraphs, in which Feynman recall when, as a student of the Princeton physics department, he used to challenge the students of the math department.
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?"
"No holes."
"Impossible! There ain't no such a thing."
"Ha! We got him! 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.
Forgetting that most of this was probably done as a joke, with what theorem would you have answered to Feynman's challenge?
Best Answer
There's a certain gaming/sporting aspect to Feynman's challenge that works in his favor. First of all, as phrased, the challenge gives him a 50/50 shot at being right even if he guesses randomly. Also, if you present a statement which seems too obviously true then Feynman could reason that you wouldn't have chosen that statement if the obvious answer were correct.
With those caveats, I present some proposals below. I have tried to gravitate toward problems involving physical intuition, since IMO fooling Feynman's physical intuition carries bonus points.
Is sphere eversion (without creasing) possible? (Edit: I see now that Noah Schweber already suggested this one.) This is my favorite, and the only flaw is that a real physical surface can't pass through itself, so the question isn't quite a physical one.
Is there a convex solid of uniform density with exactly one stable equilibrium and one unstable equilibrium? This problem postdates Feynman's life and maybe he would have guessed correctly, but I would be impressed.
Does the regular $n$-gon have the largest area among all $n$-gons with unit diameter? Tricky because the answer is yes if $n$ is odd but false if $n$ is even and $n\ge 6$.
Is there a closed planar convex shape with two equichordal points? This one is good if you suspect that Feynman might reason that the "obvious" answer (no) can't be correct or else you wouldn't be asking. "No" is correct but the proof is highly nontrivial.
Does every $d$-dimensional polytope have a realization in which all its vertices have rational coordinates? Explaining the precise statement of this result is a little tricky, and it has the flaw that the weirdness doesn't kick in until $d=4$, but otherwise this one is really nice IMO.
EDIT: There has been some discussion in the comments about whether the presence of irrational numbers in #5 above means it refers to some mathematical abstraction that is not "physically realizable." Here is an easier-to-understand (though less spectacular) question that captures the essential issue. Consider the Perles configuration of nine green points in the figure below. If the outer pentagon is a mathematically perfect regular pentagon, then at least some of the nine points have to have irrational coordinates. No big surprise there.
But now consider this. Suppose I don't require that the pentagon be a perfectly regular pentagon; suppose I allow you to redraw the figure, re-positioning the nine points in any way you like, as long as you preserve all the collinearities (i.e., points that are collinear in the original diagram must remain collinear in your new picture). Can you arrange for all nine points to have rational coordinates? The answer, which I find surprising, is no.
As far as physical intuition is concerned, what this example shows is that the seemingly physical concept of "collinearity of points in the plane" (suitably idealized of course) already forces irrational numbers upon us. In contrast, if you point out that a right-angled triangle with two sides of length 1 has a hypotenuse of $\sqrt{2}$, then a physicist could object that physical angles and lengths are never infinitely precise, and that you can create an arbitrarily close approximation using rational numbers. In the Perles configuration, we are not stipulating any distances or angles precisely; we are only stipulating that certain points be exactly collinear, and it turns out that this idealization already requires us to introduce irrational numbers.