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Paradox of Enchantress and Witch

Part 1
      Enchantress's scheme
      Witch's scheme

Part 2

Part One

Paradox of enchantress and witch is a paradox involving infinities. It can be seen as an example that is undefined operation. It is possible to get any wanted result.

A nice girl named Blondy had some financial difficulties. A kind enchantress decided to help her. The enchantress appeared in front of the girl and promised to give her infinite quantity of gold coins. The kind sorceress wanted to give money in a particular order, and numerated coins from 1 till infinity. She explained her scheme to Blondy.

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Enchantress's scheme

One minute before noon the kind enchantress will give coins with numbers from 1 till 10.

1/2 minute before noon she will give coins with numbers from 11 till 20.

1/3 minute before noon she will give coins with numbers from 21 till 30.

And so on.

At noon Blondy will have an infinite quantity of coins.

The girl was carried away by the grandiose perspective; and the enchantress disappeared.

Castles in the air were broken by an appearance of a malicious witch. A greedy witch offered a deal to the girl. She promised to increase tenfold coins, which Blondy would have at noon, if girl agrees that the witch takes 1 coin of each 10 coins given by a kind enchantress in each moment.

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Witch's scheme

One minute before noon the kind enchantress gives coins with numbers from 1 till 10 and the witch takes a coin with number 1.

1/2 minute before noon the enchantress gives coins with numbers from 11 till 20 and the witch takes a coin with number 2.

1/3 minute before noon the enchantress gives coins with numbers from 21 till 30 and the witch takes a coin with number 3.

And so on.

At noon the witch will increase tenfold girl's part of enchantress's coins.

The girl decided that it was a good bargain for her and agreed.

How much coins the girl will have at noon?
Answer: zero.

Who will have coins at noon?
Answer: witch.

Explanation

pairing numbers

Let us look at the first line. P1 ... P10 are coins with numbers from 1 till 10. The kind enchantress gave them one minute before noon. The witch took a coin with number P1.

The second line shows what happened 1/2 minute before noon. The enchantress gave coins with numbers P11 ... P20 and the witch took a coin with number P2.

We can consider a coin with any number, for example with number N. It will be taken back at N-th operation.

It is an "empty" set, and from mathematical point of view there is no paradox that the girl has nothing at noon. The witch increase tenfold girl's part of enchantress's coins at noon, but it is still zero.
10 x 0 = 0.

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Part Two

A kind enchantress was extremely kind. Looking how her coins moved from Blondy to witch she decided to help Blondy again. The enchantress said she would change witch's scheme.

One minute before noon the kind enchantress gives coins with numbers from 1 till 10 and the witch takes the first of them. It will be a coin with number 1.

1/2 minute before noon the enchantress gives coins with numbers from 11 till 20 and the witch again takes the first of them. It will be a coin with number 11.

1/3 minute before noon the enchantress gives coins with numbers from 21 till 30 and the witch takes the first of them, a coin with number 21.

How much coins the girl will have at noon?
Answer: infinity.
She will have all coins except coins with numbers 1, 11, 21, ... .
At noon the witch will increase tenfold girl's part of enchantress's coins. But it is still infinity.

How much coins the witch will have at noon?
Answer: infinity. She will have coins with numbers 1, 11, 21, ... and so on.

by Tetyana Butler
Paradox is invented by Tetyana Butler

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Complex analysis studies the most unexpected, surprising, even paradoxical ideas in mathematics. The familiar rules of math of real numbers may break down when applied to complex numbers.
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