The Riddle: We assume there is an infinite sequence of boxes, numbered 0,1,2,…. Each box contains a real number. No hypothesis is made on how the real numbers are chosen. You are a team of 100 mathematicians, and the challenge is the following: each mathematician can open as many boxes as he wants, even infinitely many, but then he has to guess the content of a box he has not opened. Then all boxes are closed, and the next mathematician can play. There is no communication between mathematicians after the game has started, but they can agree on a strategy beforehand. You have to devise a strategy such that at most one mathematician fails. Axiom of choice is allowed.
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The Modification: I would find the riddle even more puzzling if instead of 100 mathematicians, there was just one, who has to open the boxes he wants and then guess the content of a closed box. He can choose randomly a number i between 0 and 99, and play the role of mathematician number i. In fact, he can first choose any bound N instead of 100, and then play the game, with only probability 1/N to be wrong. In this context, does it make sense to say "guess the content of a box with arbitrarily high probability"? I think it is ok, because the only probability measure we need is uniform probability on {0,1,…,N?1}, but other people argue it's not ok, because we would need to define a measure on sequences, and moreover axiom of choice messes everything up. (引用終り) 以上
