whizzball1

Myriad Reality

How on earth do you all rp in this?

@User

How do you mean?

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The question makes no sense to me as a hotel question.

@Host

Well, some people understand things more easily with different ways. The hotel analogy makes it easier for me; the mathematical expression makes it easier for you.

Myriad Reality

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But it does. The entire situation could be abbreviated into a few math terms.

@Host

Maybe it does to you. But if I had received the paradox like that, it would have taken me considerably longer to figure it out. However, the idea that there's a hotel with an infinite number of rooms and all of them are filled is much easier to chew on than a function is, at least once you get into trying to add people to the hotel in infinite quantities.

Myriad Reality

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Then they should remove the fluff!!

@Host

What fluff? It's easier to think about if it's expressed in real life terms. The mathematical expression I gave is too confusing and doesn't impart the true nature of the paradox.

Myriad Reality

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That's not even a puzzle. That's a mathematical system problem with fluff all around it.

@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!

Myriad Reality

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What's the point of puzzles if they realistically make no sense on a deeper level?

@Host

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!).

Myriad Reality

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That situation could be applied in a more mathematical way. That's not realistic.

@Host

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?

Myriad Reality

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Never heard that analogy before.

@Host

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.

Myriad Reality

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.

Myriad Reality

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I'm in Spanish, so I'll be in and out.

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First of all, hot is the best. Second of all, YOU LIVE IN CALIFORNIA. IT'S HOTTER THERE.

Did you read that on today's Prophecy Watch?

@Host

It was on a news website.

Myriad Reality

Huh. Half of the Presidential candidates are opposed to gay marriage. But only one is vocal about it, and of course it's the candidate from Texas, because Texas is awesome.

Well, except for the fact that it's really hot. But otherwise, it's awesome.