A while back I shared a brain teaser that seemingly can create 13 men out of 12 by shuffling the body parts. The trick to unlocking this mystery is to notice that the picture with the 13 men has each man missing a different horizontal slice of a complete body.

12 men ↔ 13 men

Perhaps another picture would illustrate the idea better:

12 rows ↔ 13 rows

On the left are 12 rows each with 12 boxes. By shifting down the cyan boxes, 13 full rows are seemingly created but with each row having 1 box less than the original row. **Note that the total number of colored boxes is preserved.** This is the principle of equivalent exchange: you cannot create something out of nothing. Ok, now I’m just doing f***ing commercial for FMA. :p

While this puzzle is merely an optical illusion, there does exist a mathematical theorem that truly violates the principle of equivalent exchange:

Banach–Tarski paradox.

It states, in particular, that you can take a ball, separate it into 5 different parts, then rearrange them into two balls each having the same volume as the original ball—effectively doubling the original ball.

This is no optical illusion. No stupid trick either. It is a rigorously proven mathematical fact. Can I show you the 5 parts? Not really. Mathematicians can only prove that such division of the ball must exists, but they can probably never be able to produce an example.

Now you must have two questions: *Why must such division exist?* and *why can’t mathematicians produce an example of the 5 parts?*

Such division must exist because of another innocuous mathematical statement:

Axiom of Choice.

It says that from an infinite number of nonempty bags of marbles it is always possible to draw a marble from each bags. The wikipedia pages are there if you want to see the connection between the axiom of choice and the Banach-Tarski paradox.

As to the second question, those 5 parts can probably be never demonstrated, because, as it turns out, the axiom of choice can be accepted or rejected independently from mathematics and yet we can still produce consistent mathematics either way. Again, the wikipedia pages are there if you are curious exactly what that means.

So, it all comes down to the problem of choice. Is there such as thing as free will or is everything predestined? If you accept free will, then the principle of equivalent exchange is violated. But if you reject free will, then what are you living for? Hey, I think I just gave the FMA writers new materials to go another 100 chapters.

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