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.
|
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.
|
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
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.
Top
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
