Hilbert Hotel Question?

Question

Hilbert's hotel has an infinite number of rooms and is currently empty. An infinite number of busses, each with an infinite number of passengers pulls up to the hotel. Each passenger wants their own room. How can you arrange the passengers so that each one gets their own room?

(by infinite number, I'm referring to alef-null, just as a technical note)

Answer 1

Make a really long list of the room numbers (since it is infinite, this is impossible, but you can still order them). For the first bus, place each person in every other room. For the second bus, place each person in every other room that is empty. For the third bus place the people in every other room that is still left. Continue for all the buses.


In response to the critique:

I does not work for only 2 buses:
The first bus gets all the odd numbered rooms.
That leaves the even number rooms for the rest.
The second bus gets all the rooms of the form 4n+2.
That leaves all the rooms of the form 4n for the rest.
The third bus gets all the rooms of the form 8n+4.
That leaves all of the rooms of the form 8n for the rest.


This continues for ever. It seems like each bus gets half as many rooms as the previous, but since we are dealing with infinity, they really aren't.

In general: The mth bus gets rooms of the form (2^(m-1))(2n+1)

Answer 2

I was thinking of something similar to scm_abc's answer.

Number the buses 1, 2, 3, .... and so on.

Give room 1 to a person on bus 1.

Give rooms 2 and 3 from one person on bus 1 and one from bus 2.

Give rooms 4, 5, and 6 to a person from buses 1, 2, and 3.

Give rooms 7, 8, 9, and 10 to a person from buses 1, 2, 3, and 4.

And so on...

This is possible because a set consisting of a countable (possibly infinite) number of countable (possibly infinite) sets is countable.

edit: OK, eulercrosser, I misunderstood what you were saying.

rakneen's answer only works if there is an infinite number of floors with an infinite number of rooms. It is possible that there is only one or a finite number of floors with an infinite number of rooms, or that there are some floors with a finite number of floors, for example. Since nothing is stated in the problem about this, then we can't assume anything about the number of floors or how many floors there are.

Answer 3

List the passengers on a square table by putting the passengers' names of the first bus on the first row, second bus on the second row, etc. Then assign rooms to the passengers by starting from the top left corner of the table along diagonal lines from top right to bottom left.
This way the Nth passenger on the Mth bus is assigned to the [(M+N-1)(M+N)/2 - N + 1]th room, and from the above we can see that each one will get their own room.

Answer 4

since the infinite number of rooms is countably infinite, you should be able to construct a one-to-one correspondence with the rational numbers. This means that we could assume (without loss of generality-by the one-to-one correspondence of the rational numbers with any countably infinite set) that there are countably infinite number of floors in the hotel (the floors correspond to the denominator of the rational numbers), each with a countably infinite number of rooms. So each bus is assigned to one of the floors and each of the passengers will have a room by themselves on that floor.
(Edited)

Answer 5

put one in each room