"No" is correct but the proof is highly nontrivial.ĭoes 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. 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. 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.ĭoes 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 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. I have tried to gravitate toward problems involving physical intuition, since IMO fooling Feynman's physical intuition carries bonus points. With those caveats, I present some proposals below. 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. First of all, as phrased, the challenge gives him a 50/50 shot at being right even if he guesses randomly. There's a certain gaming/sporting aspect to Feynman's challenge that works in his favor.
If I guessed it wrong, there was always something I could find in their simplification that they left out.įorgetting that most of this was probably done as a joke, with what theorem would you have answered to Feynman's challenge? "No, you said an orange, so I assumed that you meant a real orange." "But we have the condition of continuity: We can keep on cutting!" 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." "Ha! We got him! Everybody gather around! It's So-and-so's theorem of immeasurable measure!" "Impossible! There ain't no such a thing."
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.
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." 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.