Infinite Hotel Paradox

September 29, 2023

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Suppose you have a hotel with infinitely many occupied rooms. Now another guest shows up, what would you do? It turns out that you can still accommodate him. In fact, you can actually accommodate a bus of infinitely many guests and even infinite busses of infinitely many guests. However, not any amount of guests can be fit.

Finite number of guests

First we will look at the scenario where one new guest arrives at the hotel. You want to allocate a room to this person but every room is occupied. This can actually be solved pretty easily: simply tell every guest to move to the next room, now room 1 is free for the new guest. The same can be done for larger groups, simply let all the guests move the same number of rooms as the new amount of guests. This way any finite number of guests can be accommodated.

Infinite Number of Guests

Now what about an infinite number of guests? This is also easier than it sounds: simply let all guests move to the room which is twice their room number. Now all even numbered rooms are occupied and there are an infinite amount of odd numbered rooms available. This way you can even accommodate an infinite number of guests.

Mathematical Concept

Let us now look at the mathematical concept behind this. Number the guests by their initial room numbers i \in {1, 2, …n}, where n lim inf. When there are finite number x of new guests, all the current guests move to room number i + x. Now the rooms {1, 2,… x} are vacated for the new guests. For an infinite number of new guests all the current guests move to room number 2i. This means all the rooms 2i – 1 are vacated for the new guests. This mathematical interpretation helps to better understand more complex situations.

The case for an infinite amount of infinitely many guests can be solved in multiple ways, the easiest one is by looking at prime numbers. Number the guests on the busses in the same way as the guests in the hotel: i \in {1, 2, …,n}. Euclid proved there are an infinite amount of prime numbers, so we can number the busses by prime numbers j \in {2, 3, 5, …} where the hotel gets the first prime number 2. Now tell the guests to move to room number j^i. As these are all powers of prime numbers, none of them will have the same room.

Larger Infinities

You might now be thinking that any amount of guests can be fit in the hotel, this is not the case however. Suppose another infinite number of guests show up. This time they all have an infinitely long name consisting of any letter in the alphabet. Furthermore, each possible name is taken. Now suppose you start assigning rooms to each of them, after a while you find out there are always guests that are not in a room. Take the first guest, pick the first letter in their name and change it. This means that any guest starting with this letter has a different name than the guest in room 1. Now pick the second letter from the second guest and change it. Now add this letter to the changed letter you picked from the first guest. Repeat this process for all the other rooms. When this is finished you are guaranteed to have a guest who does not have a room yet. This process can be repeated infinitely many times however there are always guests which do not have a room yet. This means that unfortunately you can not fit all the guests into a room.

Conclusion

This paradox first published by David Hilbert in 1924 shows the complexity of infinity. Furthermore it shows that not all infinities are equal and there are multiple sizes of infinity. There are many more examples of infinite amounts of guests that can or cannot be mapped to rooms such that all new guests can fit. In general any amount of guests can be fit as long as you can find a bijective function mapping the guests to their rooms.

After reading this article I hope you know what to do when managing an infinitely large hotel receiving different sizes of infinitely many guests. However, there is always more to think about when looking at infinity. If you are interested in the concept I definitely recommend reading the articles below.

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