Xander, I watch lot of math videos, too. I think I saw the same one at around the same time you did. I will excuse myself from this one, but it is astonishing just how good the estimate is. Billion decimal place, or some such.

18 trillion trillion decimal places

Like I said:"some such".

https://www.youtube.com/watch?v=LKvjIsyYng8

Didn't watch video. I know that puzzle, have seen the solution many times, and the answer is: No, I cannot solve it. Way too ****ing hard.

There are lots of other good videos on youtube.

Alice secretly picks two different real numbers by an unknown process and puts them in two (abstract) envelopes. Bob chooses one of the two envelopes randomly (with a fair coin toss), and shows you the number in that envelope. You must now guess whether the number in the other, closed envelope is larger or smaller than the one you’ve seen.

Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?

Check the weight of each envelope

It would be nice to be know the range of numbers, but even without it you should be able to gain a very, very slim edge by always saying the number in the other envelope is larger. No matter what number Bob discovers (even 1,345,667,885, for example), there will always be a fixed number of choices less than that number, but an infinite number of options higher than the number.

The good old problem with infinity. As soon as you see the a whole number in the envelope. There is a 100% chance the other envelope is larger. Which is why mathematicians are careful with infinity.

it doesn't say that the number has to be positive so there are infinitely many numbers smaller than it too

Infinity + all the positive numbers lower than the first, so infinity +....lol

so if the number is positive then you pick lower and if the number is negative then you pick higher

No offense but you all are missing the completely obvious answer here:

It does not specify that Alice uses a uniform distribution between the infinities, so that strategy doesn't work

I must be missing something, because the only thing I can think of is "No of course not. That's stupid."

But why else would you be asking the question?

This.

Does Alice always give out her number this way?

Is there some information missing in this puzzle?

As it's written, I'm in the "no" camp and can't imagine an explanation that would change my mind.

As I understand the question (and please correct me if describing a non-equivalent situation):

I put a number in a yellow envelope (we'll call this Y). I put a different number in a blue envelope (we'll call this B). I selected these numbers by any manner I felt like. Maybe I rolled a couple dice. Maybe I asked two other people. Maybe I got real lazy and just selected 1 and 2.

Now that the envelopes are sealed, one of two options is true:
B>Y
Y>B
Either of these options is equally likely.

You randomly (or not randomly, I don't see why it matters) select the yellow envelope and see what the value for Y is.

I don't see how this helps. It could be 1 or negative 40 octillion or Graham's number. But without knowing ANYTHING about the metric I used to select the numbers, I don't see how this offers any new information.

Is there some cute wordplay trap I'm missing?

I think that's everyone's first reaction, but the solution is actually quite nice and seems almost obvious in hindsight

Did my last post describe an equivalent situation?

You have the technical description correct. There's no word play and it's asking exactly what you think it's asking.

I'll post a first hint tonight

I will say that it's important that you get to select your first envelope, it's not the same one every time, otherwise a malicious Alice would be able to beat you.

I think I'm more confused.

The only possible real answer I can think of:

Go ask Alice, when she's ten feet tall.