Recently I chewed the fat with a physics student and got intrigued by him mentioning "the Devil's problem," which he described as a simply worded mechanics problem that is extremely difficult to solve and has an answer of exactly 13 despite the formulation having no numbers and being very natural. That's kinda crazy, so I laughed and said he was kidding, but he responded that no, he wasn't kidding.
He then explained me the formulation of the problem. You've got a plastic tube, like a tube used to carry posters to conferences, but open on both ends. You firmly attach a thin but heavy rod to the inner surface of the tube, parallel to the tube's axis. The tube is then laid on a floor so that the rod is in the uppermost position, and then is released to roll away from that unstable equilibrium position. The floor isn't slippery, so there's no sliding. How many times heavier than the original tube must the rod be for the tube to jump?
I'm a student studying something unrelated to physics, and although I liked physics at school, this problem is too tough for me to crack, so I can't tell whether he was fooling me or whether what he said is true. I tried to find the problem on the Internet, but to no avail, so I'm posting it here. It's mystical and a bit scary if the Devil's dozen really pops out of nowhere in such a simply stated problem, but I guess the student was bluffing, counting on my inability to solve such problems. I don't even understand why the tube would jump.
Can the rod actually jump? If so, how can one approach this problem? Can you help me call the bluff of the student, or is 13 really the answer?
UPDATE: To clarify in response to a comment below, the rod and the wall of the tube are much thinner than the tube diameter and thus can be assumed to be infinitesimally thin. Likewise, an infinitesimal initial perturbation due to a slight asymmetry or thermal fluctuations is assumed. The problem is clearly well-posed from the mathematical standpoint, so the only question is how to solve it and what is the answer.
UPDATE 2. It seems I've figured out why the tube will jump if the mass of the rod is large enough, but I can't calculate the exact threshold. My proof of the jump is in my answer below.