Newcomb's paradox is one of the most simply stated but astonishing of the so-called prediction paradoxes that bear on the problem of free will. It was devised in 1960 by William Newcomb, a theoretical physicist at the Lawrence Livermore Laboratory and the great-grandson of the brother of the astronomer Simon Newcomb, while contemplating the prisoner's dilemma.
A superior being, with super-predictive powers that have never been known to fail, has put $1,000 in box A and either nothing or $1 million in box B. The being presents you with a choice: (1) open box B only, or (2) open both box A and B. The being has put money in box B only if it predicted you will choose option (1). The being put nothing in box B if it predicted you will do anything other than choose option (1) (including choosing option (2), flipping a coin, etc.). The question is, what should you do to maximize your winnings? You might argue that since your choice now can't alter the contents of the boxes you may as well open them both and take whatever's there. This seems reasonable until you bear in mind that the being has never been known to predict wrongly. In other words, in some peculiar way, your mental state is highly correlated with contents of the box: your choice is linked to the probability that there is money in box B. These arguments and many others have been put forward in favor of either choice. The fact is there is no known "right" answer, despite the concerted attentions of many philosophers and mathematicians over several decades.
1. Gardner, M. Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, 1986.