Franz Kiekeben (who is a very funny cartoonist) does a nice job of describing Newcomb's Paradox, which I've enjoyed contemplating, on and off, for many years.
A highly superior being from another part of the galaxy presents you with two boxes, one open and one closed. In the open box there is a thousand-dollar bill. In the closed box there is either one million dollars or there is nothing. You are to choose between taking both boxes or taking the closed box only. But there's a catch.
The being claims that he is able to predict what any human being will decide to do. If he predicted you would take only the closed box, then he placed a million dollars in it. But if he predicted you would take both boxes, he left the closed box empty. Furthermore, he has run this experiment with 999 people before, and has been right every time.
What do you do?
On the one hand, the evidence is fairly obvious that if you choose to take only the closed box you will get one million dollars, whereas if you take both boxes you get only a measly thousand. You'd be stupid to take both boxes.
On the other hand, at the time you make your decision, the closed box already is empty or else contains a million dollars. Either way, if you take both boxes you get a thousand dollars more than if you take the closed box only.