There is a hotel with an infinite number of rooms, and on one particular night, all its rooms are occupied with one guest. On that night, a guest shows up. How can we accommodate this guest, when the hotel is full? We can ask our guests to move, but this must take a finite amount of time, and there must be only one guest to a room. There is more than one answer.
The answer (one answer) is spoiler'd for those who haven't seen this problem, and for organization.
In another words, we supply every current guest with a function, in this case, f(n)=n+1. Each guest moves to room f(n), where n is their room number. We give the new guest the number 1 (the first room).
That was the easy part (one of them). Now, let's say that just as our guests are finished unpacking their things in their new rooms, a bus with an infinite number of passangers pulls up to the hotel. The question is, using the same rules as before, how do we supply all the passangers on the bus with a room?
The last part is the same except instead of one bus, there's an infinite number of buses.
My question is why stop there? What if an infinite number of (rows?, columns?, caravans?), each containing an infinite number of buses, each containing an infinite number of passangers, shows up at the hotel?
I guess the problem is to come up with an injective (not necessarily bijective) function from N(all natural numbers, starting with 1, I suppose)^3 to N, where the last problem was to come up with an injective function from N^2 to N.
Obviously (I think), the next step would be to come up with an injective function from N^4 to N, and so on ad infinitum. How about N^n onto N? Then what (assuming the steps up to this point are even possible)?
EDIT: So, for now, can anyone come up with a solution to the fourth part? Also, is there a bijective solution to part three (like part one and two, leaving no empty rooms)?
EDITEDIT: Would this be better off in the Mathematics forum?