One Observation Behind Two-Envelope Puzzles

In two famous and popular puzzles a participant is required to compare two numbers of which she is shown only one. The first puzzle is known as the paradox of the double sum, the second, due to Cover is known as "which is the larger number". Although the puzzles have been discussed and explained extensively, no connection between them has been established in the literature. We show here that there is one simple principle behind these puzzles. In particular, this principle sheds new light on the paradoxical nature of the first puzzle. According to this principle the ranking of several random variables must depend on at least one of them, except for the trivial case where the ranking is constant. Thus, in the nontrivial case there must be at least one variable the observation of which conveys information about the ranking