The paradox is that, since “when I see an amount, the other envelope will contain twice my amount half the time and half my amount half the time” is true, therefore trading every time will win on average, when, in fact, trading every time won’t win on average. The paradox is this contradiction.

To solve the paradox, I must show that the belief that “when I see an amount, the other envelope will contain twice my amount half the time and half my amount half the time” is false. That way, the conclusion that trading *every time* will win on average cannot be drawn, the contradiction is removed, and the paradox resolves.

The belief that “half the time I’ll lose half the amount and half the time I’ll win twice the amount” implies that ***that outcome is always true for every amount chosen***. If the quoted logic fails for any amount, the logic is deemed false, which makes trading every time suspect, which removes the contradiction and, therefore, the paradox.

In fact, that outcome is ***never true for every amount chosen*** because it is never true for the largest amount put into whatever set of envelopes is chosen.

I think that the above paragraphs successfully resolve the paradox.

In your paired example, the logic always fails for the largest amount ([2^k]n at k-max).

For a moment I considered that the only amount in your sequential pairs example that loses money on average is the largest amount. That means that the odds of picking an amount that wins money is greater than the odds of picking an amount that loses money, and the paradox still holds. While the logic is correct as far as it goes, it misses a necessary piece of information: the amount lost in that sole losing choice is equal to the winning amount for all the other amounts combined. Thus, the (weighted) average of losing and winning is still zero, and the paradox doesn’t hold.

If someone says, “I’ll win half the amount half the time and twice the amount the other half of the time,” I’ll say, “That’s not true with every amount that you can choose.” That stops the logic from concluding that, “I’ll win on average by trading *every time*,” which stops the paradox. For example, if she sees 20D and says, “half the time I’ll win 20D and half the time I’ll lose 10D,” I’ll say, “That’s not true if the person has decided to never use 40D.” She might say, “But I think he’s likely to use 40D, so I’ll trade every time with this amount.” Her word “likely” clearly demonstrates her understanding that trading every time *might lose* (but she thinks it is unlikely) which then stops the logic from being definitely concluded, which resolves the paradox. She is demonstrating a different strategy that I have previously described.

Your wrote, “Some philosophers say that the solution to the puzzle is getting the player to take this viewpoint and ignore everything else as illusory.” I agree. For me, the problem with this approach is that it doesn’t address the *reason* that the paradox is wrong. It just shows that the conclusion is wrong, but not why. For me, this is no small issue. At different times during my education, I approached professors with questions about a problem. I’d explain my logic and they’d explain the correct logic. Sometimes, the correct logic didn’t explain the problem with my logic. Their approach taught me how to approach the problem correctly, but I didn’t learn what was wrong with my logic. I might ask, but they’d often just defer to the correct answer. When I was able use the correct answer to eventually understand why my logic was incorrect, I then understood the problem and the correct logic much better, and in the future, with different but similar problems that were more complex, I’d avoid the same faulty logic. In fairness, explaining the correct logic is much easier and more time efficient, and trying to understand the twisted logic that sometimes arises when people are learning new and complex math can be very difficult and time consuming. So, I understand why I sometimes was told the right answer without addressing my logic. Still, I think that effort put into explaining faulty logic (and not just giving the correct answer) can have significant benefits.