You are told there are two envelopes. One contains twice as much money as the other one. You pick one but are allowed to change your mind after picking it. (You are equally likely to pick the one with less money as the one with more money.)
To figure out how much on average the other envelope should contain, one might average x/2 and 2x because one is equally likely to pick one as picking the other. Since this comes out to 5x/4, one might always change his or her mind. But wouldn't this end up with one never making up his or her mind?
What Gamer has done is tricked us by assigning x two different values. Assume for a moment one envelope contains $3 and one contains $6. Let's make x the $3. If we average x/2 and 2x, we are averaging $6 and $1.5. Aha! these are not the original values.
So let's assume x is $6. Averaging x/2 and 2x would be averaging $12 and $3. Aha! these are not the original values either.
It should be averaging x and x/2, or x and 2x. But great paradox, Gamer. You got me thinking.