## The two envelopes puzzle.

This week, Tim Urban posted an interesting mathematical paradox on his blog, *Wait But Why*. Urban sets up three situations, each building on the next, that all seem reasonably sensible until you get to the end, and suddenly nothing makes sense.

*SEE ALSO: The Sleeping Beauty Puzzle Has Two Contradictory Correct Answers*

In situation 1, a friend offers you $20. He then gives you the option of keeping the bill or trading it in for an envelope that contains either $10 or $40. What’s your best move? Well if we were to repeat this activity twice, there were a fifty-fifty chance of the envelope containing each amount, and you stuck with the $20 both times, you’d end up with $40. However, if you chose the envelope both times, you’d have $50. Thus, you’re better off choosing to go with the envelope.

The second situation is very similar: There are two envelopes with identical appearances. You open the first one and it contains $20. Your friend tells you that one envelope contains double the amount of the other. Since you don’t know which envelope is which, this situation is identical to the first situation—you have to choose whether to stick with $20 or trade it in for an unknown amount that will be either $10 or $40. As before, you’re best to take the second envelope.

In situation 3, things get a little more abstract. As in situation 2, one envelope has double the amount the other has, but this time, you don’t get to open the first envelope before making your choice about whether or not to trade it in for the second envelope.

We can do the same math as in situation 1 by calling the amount in the first envelope x. The second envelope must have either 2x or 0.5x in it. As before, if you follow the math, you’re better off switching envelopes. If you repeat the experiment and keep the first envelope both times, you get twice the amount in it — 2x — but if you switch envelopes both times, you get 0.5x once and 2x once, so you end up with 2.5x in total.

However, your friend now throws a wrench in things by asking whether you want to switch envelopes again. Since you still haven’t opened either envelope, it looks like the best thing to do is say yes. However, your friend who apparently likes to torment you says that you can change your mind as many times as you’d like.

You end up in a vicious circle, endlessly switching envelopes and never get any money. What just happened? Can you find the flaw in the reasoning? Let us know in the comments!