A different (maybe better) title for this might be: the most frequently banned topics on the internet. Below I present a set of questions that are so counter-intuitive that their very mention can set off the right audience into a complete flame war. Each of them has been known to paralyze various online communities resulting in monster threads and lots of insults (and in one case, the company stepping in to solve the dispute). You can tell the intellectual average of a community based upon the sophistication of the brain-exploding questions that give it fits.
Is .9999~ = 1 or is .9999~ < 1?
When I was 12, this problem ruined my world. The answer is, of course, .9~ is exactly equal to 1. In elementary school, they tried to placate me with algebraic trickery, but I wasn't buying that. Obviously, .9~ is the number right before 1, but not equal to 1. Obviously! It took me awhile to come to terms with the fact that any two distinct real numbers have an infinite amount of numbers between. If .9~ and 1 were distinct, they'd have an infinite amount of numbers between them. Since they don't, they cannot be distinct. And so I moved on.
Suppose you’re on a game show and you’re given the choice of three doors. Behind one door is a car; behind the others, goats. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to change your choice?
When I was in high school, this problem ruined my world. The answer is, of course, you should switch doors. At first, you might believe, that you are still choosing between two doors, and so it doesn't matter, it's 50-50. After you think a little bit more carefully you realize that the rules that Monty plays by gives him no choice but to give you information, making the choices not so equal. Intuition squares with this, eventually.
A plane is standing on a large treadmill or conveyor belt. The plane moves in one direction, while the conveyor moves in the opposite direction. This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction). Can the plane take off?
When I was in college, this problem ruined my world. The answer is, of course, the plane will have no trouble taking off. Alright, that's not quite true. The problem with this problem is that it's open to a bit of interpretation. A more precise answer is if the pilot wants to take off, he will, regardless of what the treadmill does. This is basically because the airplane wheels are not where the power is, the engines are. And the engines are pushing against the stationary air, not the moving treadmill.
There are two games of bridge going on somewhere in the world, right now. In one of those games, Alice announces, "I have an ace in my hand." In the other of those games, Bob announces, "I have the ace of spades in my hand." Whose hand is more likely to contain a second ace, Alice or Bob?
In grad school, this problem ruined my world. The answer is, of course, Bob is much more likely than Alice to have a second Ace. I've run the math on this and it's absolutely true. I still don't have a good intuition for why.
You are given two envelopes, one of which contains twice as much money as the other. You asked to choose one to keep. No matter which envelope you choose, the other one has a higher expected value. So you should switch your choice. But now... the original has a higher expected value.. so you should switch back. But now...
This problem, and its variants, still ruin my world. The answer is, of course, well, uhm. I actually heard this problem in college and not only do I not have a good answer to what's going on, apparently no one does. That makes me feel better.
In the simple version above, the flaw in the logic comes at the very beginning where it is presumed that the initial value is an unbounded and uniform random number. An unbounded, uniform distribution is impossible, and so the number chosen cannot be from that distribution. Certain possible distributions have solutions. However, there are variants of this problem that have possible distributions and still present the paradox. Good luck with that.