Here are two envelopes. One contains twice as much money as the other. You must choose one, and then consider whether to keep it or exchange it for mine. Which should you do?

It would seem advantageous to switch: Depending on which envelope you started with, you’ll either lose a little or gain a lot. (If your unopened envelope contains $10, for example, the other must contain $5 or $20.)

So we trade envelopes and I offer you the same deal. But now the same reasoning applies, so it makes sense to trade again. Indeed, it seems reasonable to keep exchanging envelopes forever, without ever opening one. How can this be?