Paradox of wizard and mermaid

Paradox of rich wizard and mermaid is a paradox involving infinities. It can be seen as an example that

is undefined operation. Playing around undefined operation it is possible to get any wanted result.

An infinitely rich wizard has a mermaid in the pond of his garden. He likes to play with her and gold coins. Every minute he throws two coins in the pond and she throws him back one coin after a half minute.

When this game is allowed to run forever, how will the money be distributed? We are allowed put a label on all the coins, numbering them 1,2,3,4, etc.

Solution 1

The wizard gives the mermaid coin number 1 and 2, she returns number 1. He gives her number 3 and 4, she returns number 2. He gives her number 5 and 6, she returns number 3. And so forth.
Conclusion: eventually each coin returns to the wizard, so he will have everything back again and she has nothing.

Solution 2

The wizard gives the mermaid coin number 1 and 2, she returns number 2, keeping number 1. He gives her number 2 again together with 3, she returns number 3, keeping number 2. He throws number 3 and 4 in the pond, she gives him number 4 back, keeping number 3. And so forth.
Conclusion: at the end she will keep all the coins, and the wizard has nothing.

Solution 3

The wizard gives the mermaid coin number 1 and 2, she returns number 2, keeping number 1. Next time, however, the wizard throws coin number 3 and 4 in the pond, the mermaid keeping number 3, returning number 4. And so forth.
Conclusion: at the end she will have all the odd numbered coins and the wizard all the even numbered, so both are equally rich.

by Tetyana Butler