Here is a game that looks harmless and hides a trap. Someone shows you two sealed envelopes and tells you one contains exactly twice as much money as the other. You pick one at random and hold it. Before you open it, you are offered a swap. Should you take it?
An argument for switching sounds airtight. Say your envelope holds some amount, call it A. The other envelope is equally likely to hold twice as much or half as much, so it contains either 2A or A/2. Average those two outcomes and you get 1.25A, more than the A in your hand. So you should switch. The trouble is that once you switch, the same reasoning applies to the new envelope, telling you to switch back, and then forward again, forever. You are stuck in a loop, endlessly wanting the envelope you are not holding.
Something has clearly gone wrong, because two identical envelopes cannot each be worth more than the other. The flaw is buried in that innocent letter A.
When you write “the other holds 2A or A/2”, you are quietly using A to mean two different things. In the case where you are holding the smaller envelope, A is the small amount and the other holds 2A. In the case where you are holding the larger one, A is the big amount and the other holds A/2. Those are two different situations with two different values of A, and the calculation smashes them together as if A were a single fixed number. Once you track the two real amounts honestly, say the envelopes hold X and 2X, the appeal of switching evaporates: you have a fifty-fifty shot at either, whether you switch or not.
The puzzle endures because the bad argument is so much more comfortable than the good one. Our minds love a clean expected-value sum, and 1.25A feels like a proof. It takes a second look to notice that the symbol quietly changed its meaning halfway through, which is exactly the kind of move that turns arithmetic into a magic trick.
