I call my choice of envelope x.
Then the other must be 2x or 1/2x.
I like the sound of that, so I swap.
So my first choice is now the other envelope and is 2(2x or 1/2x) or 1/2(2x or 1/2x) = 4x or x or x or 1/4x = 4x or x or 1/4x
I like the sound of that even better, so I swap again
So the other envelope is now 2(4x or x or 1/4x) or 1/2(4x or x or 1/4x) = 8x or 2x or 1/2x or 2x or 1/2x or 1/8x = 8x or 2x or 1/2x or 1/8x.
Yes, you got it, I swap again ... and again till the first term is like (2^100)x or something, then I open the envelope.
That's how I get to be one of those mysterious billionaires.

I call my choice of envelope x.

Then the other must be 2x or 1/2x.

I like the sound of that, so I swap.

So my first choice is now the other envelope and is 2(2x or 1/2x) or 1/2(2x or 1/2x) = 4x or x or x or 1/4x = 4x or x or 1/4x

I like the sound of that even better, so I swap again

So the other envelope is now 2(4x or x or 1/4x) or 1/2(4x or x or 1/4x) = 8x or 2x or 1/2x or 2x or 1/2x or 1/8x = 8x or 2x or 1/2x or 1/8x.

Yes, you got it, I swap again ... and again till the first term is like (2^100)x or something, then I open the envelope.

That's how I get to be one of those mysterious billionaires.