The reason this problem is controversial is because it’s TRICKY, and people hate it when they are tricked. Here is another similarly tricky problem: A couple has two children, one of whom is a boy. What is the probability that the other child is also a boy? The answer is a counterintuitive 1/3 because the problem does not state WHICH child is a boy. If the problem had instead stated, “the first child is a boy,” then the answer would be the expected 1/2.

Of course, when you explain that to people, they get mad and say, “Well, you tricked me! It’s an unfair question. You didn’t say which child was the boy, so I assumed it was the first one.” Exactly, you cannot make unfounded assumptions. That’s what the riddle is trying to teach you.

Likewise, the two envelopes problem is a lesson on symmetry. It’s TRICKY because people don’t realize that BOTH envelopes contain either double or half what’s in the other. It’s a symmetrical statement. Even if you reason yourself into switching envelopes, once you’ve switched it then turns out that my envelope STILL contains either double or half what’s in your envelope, so by your reasoning you should switch again. Which is, of course, ridiculous.

People like to complain, “It’s an unfair question. I thought you meant that my envelope was filled first with a constant value of $100, and then yours contains either $200 or $50.” Of course, that’s not what the problem says, and people are making unfounded assumptions.

Also, the incorrect solution that you are talking about, solution 1 on my site, was included on purpose specifically because it’s wrong. I was turning the riddle into one of those, “Here’s a proof that 1 + 1 = 3, where is the error?” type of problems.