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souleaterfan320
Nimda
Audacious<Snipped quote by Host>
Darn it, it's just something to think about and solve! It doesn't have to have a deeper philosophical meaning or be something you can do in the real world!
Then they should remove the fluff!!
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Then they should remove the fluff!!
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I'm good.
Good, good.
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The point is to think about them, and improve yourself figuring them out. At least, that was my goal in solving that puzzle. I'm a better thinker for it. Although, according to the Wikipedia page, the point of the paradox was "to illustrate certain counterintuitive properties of infinite sets." Namely, that even all of infinity was filled, you could still fit in another element, or you could fit in infinite new elements, or you could even add infinite groups of infinite elements, and mathematically it would actually work (with empty elements left over!).
That's not even a puzzle. That's a mathematical system problem with fluff all around it.
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It's not supposed to be realistic. It's just a paradox, something for mathematicians to think about. The maths book gives the question because it wants the reader to stop and think for a while about it. Although I suppose it can be expressed more mathematically.
Consider a function f:N->N where the domain and codomain are the natural numbers, and each element in the domain is mapped to itself in the codomain. Is there a way to map the infinite elements of each of a group of infinite sets to the elements in the codomain, in addition to mapping those in the original domain, in a 1-1 manner?
What's the point of puzzles if they realistically make no sense on a deeper level?
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How are you?
I'm okay, thank you. And you?
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Kat: Anyway, it was nice to meet you.
Virgil: Yes, It was very nice to meet you.
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Virgil: *Smiles* Great... I'll see you later then.
*She and Kat Leave the room, Virgil slightly leaning on Kat.*
Goodbye.
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Really? Well, say you had the hotel and it was filled up, as is usually the original setting. But then, in front of the hotel come an infinite number of buses, each with an infinite number of people, all of which need to go into the hotel. And so the hotel manager takes all the people from the hotel and loads them into an empty bus he has. So now you have infinite buses which need to go into the hotel, labelled 1, 2, 3, 4... How does he arrange them so that all the people get a room? (This is a bit of a variation in that the people in the hotel are removed first.)
I initially considered taking the people from bus 1, numbering them, squaring their number, and putting them in the room that was the answer. Then the people from bus 2 would be numbered, their number would be cubed, and that new number would be their room. I thought this pattern might continue to work because no natural number is both a square root and a cube root, except for 1.
Then I tried bus 3, and raised those people's numbers to the fourth power, but I realised that every single fourth power would be a perfect square, because raising a number to the fourth power is the same thing as squaring the number and then squaring that.
So, my thought process moved to trying to figure out a series of numbers that could never be the product of two previous numbers that were used. This was clearly the prime numbers. So, I considered numbering the people in bus 1, squaring their numbers, and assigning them to those rooms. The numbers of the people in bus 2 would be cubed. The numbers of the people in bus 3 would be raised to the fifth power. By this, I realised that none of these powers would ever overlap. No fifth power could be a third power. No 101th power could be a 73rd power. And thus my solution was found. Expressed algorithmically:
For each bus B, take each passenger P, raise P to the power of the Bth prime number, and assign that passenger to the answer. The answer given in the maths book was different, but I went to the Wikipedia page and found that my solution was one of those listed.
That situation could be applied in a more mathematical way. That's not realistic.
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Kat: *Helps you pick up* And thanks for that help.
I'm glad I could.
*Stores them in my pocket*
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Virgil: Yeah. I'm fine, just tired. *She gets up* Anyway. sorry to interrupt your book reading with our Drama.
No, no. It's fine.
*Walks around to pick up my darts*
I'm happy to help when I can.
Today, my maths book told me about Hilbert's Paradox, and asked me to try and think of a way to fit all the passengers from infinite buses, each having infinite passengers, into a hotel already filled with infinite people. (All those are the natural infinities.) Somehow, I managed to figure out the prime powers solution despite having never heard of it before. That was fun.
Never heard that analogy before.
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