The brain teasers were all solved by multiple commenters; I’ll summarize their answers at the end of this post. The special relativity problem proved trickier; here it is again:
A circular train (front of the locomotive attached to the rear of the caboose) sits on a circular track. At some point, the train accelerates and starts traveling around the track. Because the train is moving, I (an observer stationary relative to the track) should see it shrink. But the track doesn’t shrink. So the train can’t stay on the track, and gets pulled inward, ending up inside the track. On the other hand, the passengers say the track has shrunk, so they should expect to get pushed outside the track. How can everyone be right?
Now to the answer.
First, “At some point, the train accelerates” is ambiguous. Presumably it means that each part of the train accelerates at the same time, but of course “at the same time” means something different to a train passenger than it does to you and me (the observers stationary relative to the track).
But let’s resolve this ambiguity in the natural way by assuming that the entire train starts moving at the same time as measured by you and me. In that case, we do not see the train shrink. How could we? The front and back ends of each car have, at every moment (as measured by our watches) been moving forward at identical speeds. Given that, the distance between those front and back ends (a measured by our meter sticks) cannot change. Ditto for any couplings between the cars.
What just became of relativity? If the cars are in motion shouldn’t they appear smaller to us than to our friend Jeeter, who’s riding on the train? Sure. But that doesn’t mean we have to see Jeeter’s train car get smaller. In this case, it means that Jeeter sees his car get bigger—because by his watch, the front of his car started moving before the back did, so his train car got stretched out.
But that doesn’t mean Jeeter sees the entire train get bigger. Yes, his car got stretched when the front started moving before the back. But the car opposite him (180 degrees around the track) got shrunk when its back end started moving (according to Jeeter’s watch) before its front.
So nobody has to see the train change size and nobody has to believe the train leaves the track—which is good, because the train doesn’t leave the track.
I have ignored the fact that Jeeter is not in an inertial frame, which complicates the calculation of exactly what he experiences, but I’m nearly sure that the above captures everything important. If you want, replace the circular track with a nearly square track (with slightly rounded corners if you like) so that most of the train passengers are in inertial frames at any given moment.
Now to the brain teasers: For #1, choose a number randomly c from (say) a normal distribution on the real numbers and compare it to the number x that I’ve just revealed to you. If x is greater than c, guess that x is the larger of my numbers; if x is less than c, guess that x is the smaller of my numbers. Your chance of winning is 1 if c is between my two numbers and 1/2 otherwise; this makes your overall chance of winning greater than 1/2.
For #2, Jon Shea’s answer is perfect.
For #3, New Mexico/Colorado is one of many good answers.