Bank and mermaid
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The bank and mermaid is a paradox involving infinities. It can be seen as an example that one can sum a non-convergent series to any wanted result.
A bank director, infinitely rich, has a mermaid in the pond of his garden, infinitely avaricious. Every minute he has to throw two coins in the pond to keep her satisfied, but as a kind of penitence she will hand him back one coin every 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.
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[edit] Solution 1
The director 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. Eventually each coin returns to the director, so he will have everything back again and she has nothing.
[edit] Solution 2
The director 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 director has nothing.
[edit] Solution 3
The director gives the mermaid coin number 1 and 2, she returns number 2, keeping number 1. Next time, however, the director 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 director all the even numbered, so both are equally rich.
