Math Puzzle

This cool puzzle (and solution) comes from my friend Mike.

Alice secretly picks two different real numbers by an unknown process and puts them in two (abstract) envelopes.  Bob chooses one of the two envelopes randomly (with a fair coin toss), and shows you the number in that envelope.  You must now guess whether the number in the other, closed envelope is larger or smaller than the one you’ve seen.

Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?

I initially thought there wasn’t, or that the problem was paradoxically defined, but it turns out that it’s perfectly valid and the answer is “Yes.”  See my first comment for an example of one such winning strategy.

This puzzle is similar to, but not the same as, the two envelopes paradox.  For more time-devouring reading, see Wikipedia’s List of Paradoxes.

411 thoughts on “Math Puzzle

  1. There is of course, a really simple method of finding the answer, and that is through the amazing power of liberal arts, namely linguistics.

    The question is whether we can guess with more than 50% accuracy whether the number we chose is “smaller or larger”.

    It was also two different reals, so they cannot be the same.

    Therefore with 100% accuracy, we can say that yes, this number is smaller or larger than the other number, which is what the riddle asked about.

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