This series of blog posts, and the following ones, will explain to you some interesting properties of the Infinite Hotel.
So, first up: what is the infinite hotel?
It's simple: the infinite hotel is a hotel which, for every room, there exists another room of a larger room number. In other words, the set of all room numbers in the hotel is the same as ℤ.
So, let's take a look at our first scenario:
Not full; another person arrives
Imagine that the infinite hotel is not full, when someone arrives. To solve this is simple; you just assign the person who arrives to the smallest-numbers unoccupied room.
Full; another person arrives
This is slightly more difficult. Imagine that every room in the hotel, from 1 onwards, has a person in it, when another person arrives. Is it possible to accomodate this new person?
The answer, surprisingly, is yes. This new person can have their own room in the hotel.
The way to do this is to ask everyone in the hotel to add one to their room number, and move into that room. This means that every single room has one person leaving it, and every single room has one person entering it.
Except, of course, for room number 1.
The new arrival then goes into room 1.
Full; an infinite number of people arrive
Suddenly, a bus pulls up into the car park of the infinite hotel. This bus, however, is infinitely long, and contains an infinite number of people. Each of these people has a shirt, printed upon which is a natural number. Can they be accomodated?
To do this, everyone in the hotel must go to the room that is double their current room number. This means that all rooms that satisfy $ n = 2k-1 $ are now empty. Each person on the bus goes into a different room, where k is the number on their shirt.
In the next blog post, I will post about what happens if an infinite number of infinite buses arrives. Watch this blog for when it comes out.