[quote=@whizzball1]
<Snipped quote by 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.
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That situation could be applied in a more mathematical way. That's not realistic.