An image of a hotel with infinite rooms. The sign on top of the hotel says, "HOTEL FULL!". The stick figure of a person is standing outside the hotel with a question mark over the head.

What is so special about a math hotel with infinite rooms? For starters, it is a hotel that exists only in the mathematical world. This is because we are currently able to conjure up infinite rooms only in the mathematical world. However, what this enables us to do is to play a game with infinity. Infinity is normally a vague concept for most people, and such a game enables anyone to grasp, understand, and even play with infinity.

What game am I talking about? Imagine that you are the manager of the math hotel with infinite rooms. Given the condition that the hotel is full, can you accommodate one new guest who has shown up at your doorstep? This is where the game begins.

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Accommodating One New Guest

How can you accommodate a new guest in a hotel that is already full? Your intuition would say that this is not possible. As a result, you are likely to turn the potential guest down. But hold on! This is a math hotel we are talking about. And it is not just any math hotel; it is a math hotel with infinite rooms! You see, infinity is not a single number. It is a concept. What it denotes is that the number of rooms in the hotel is no longer countable for us as humans. All you will need to do is to ask each guest to move one room number towards the right (increase room number by one), and this would leave the first room free. You can then proceed to accommodate the new guest in the first room like so:

The sign on top of the hotel with infinite rooms now reads, “NOW OPEN!” Each person from room number 1 onwards shifts to one room number to the right. As a result, room number 1 because available for the new guest.
Image created by the author

Similarly, if many guests show up, you just need to ask each existing guest to move to a room number that is the sum of their current room number and the total number of new guests who have shown up. For example, if 13 new guests have shown up, the guest currently in room number 40 needs to move to the 53rd room (40+13). If you are still confused as to how a full hotel can still have rooms, not to worry. Just note that this is a paradox caused by invoking infinity.

How Does the Hotel with Infinite Rooms Work?

This game was originally invented by the influential German mathematician, David Hilbert. He was a key figure in the development of mathematics in the 19th and 20th centuries. Jon Von Neumann was one of his students and an assistant. Richard Courant was also one of his doctoral students among 69 others. Many of these people went on to become influential mathematicians themselves. It is safe to say that Hilbert was kind of a big deal in the mathematical scene.

An picture of David Hilbert, the renowned German mathematician who invented the game of the hotel with infinite rooms.
By Unknown author — Possibly Reid, Constance (1970) Hilbert, Berlin, Heidelberg (Image from Wikimedia Commons)

He came up with the hotel thought experiment in order to playfully demonstrate the challenges with the concept of infinity. Since infinity is not a number and does not have a range, there is no last guest in the hotel. Since the list of guests goes on beyond the countable range, moving the guests enables room for further guests. So now, the question is how many guests can we actually accommodate in the hotel?

Can the Hotel with Infinite Rooms Accommodate Infinite Guests?

What if a spaceship with infinite guests shows up at the hotel’s doorstep? Do you think you can accommodate all the new guests as the hotel manager? The answer is that you indeed can! How you do it is a little tricky. What you need to do is to ask each guest to multiply their room number by 2, and then move to the new room number that is the result of the product. Imagine being the one billionth guest; it would be a long walk! Thankfully, neither of us is a guest in this hotel. The logic with the movement is as shown in the image below:

The sign on top of the hotel with infinite rooms still reads, “NOW OPEN!” The existing guests move to new room numbers equal to twice their current room numbers. For example, the guest from room 1 moves to room 2, the guest from room 2 moves to room 4, the guest from room 3 moves to room 6, and so on. As a result, all the odd-numbered rooms become available for new guests, and they seem to be moving in.
Image created by the author

Since all of the current guests have moved to room numbers that are multiples of 2, all the odd-numbered rooms are now available. And guess what? Odd numbers stretch to infinity as well. Thus, we can now accommodate an infinite number of guests in the math hotel. It seems like a mind-boggling concept, right? But wait, the game goes on!

Infinite Spaceships with Infinite Passengers Each Show Up at The Door

We are now dealing with infinite spaceships at the doorstep. Inside each of these spaceships, we have infinite guests. What now? It is still possible to accommodate ALL of the new guests. You first need to request all of the current guests to move to their current room number multiplied by 2. This would leave all odd-numbered rooms unoccupied just like before.

Now, if n is the seat number for the passengers in spaceship 1, request each passenger to take room number 3^n. This means that the passenger from seat 1 from spaceship 1 would take room number 3, the passenger with seat 2 would take room number 9 (3²), and so on. Similarly, for spaceship 2, request the passengers to take room numbers 5^n (where n is the seat number in spaceship 2). What we are doing here is taking advantage of the fact that every whole number can be written as a product of primes in a unique way. Moving on, spaceship 3 will use the series 7^n, spaceship 4 will use the series 11^n, and so on. This way, for any spaceship number s, if n is the passenger’s seat number, we set the room p^n, where p is the (s+1)st prime number.

Where Does The Game Stop?

How many such layers of infinity can we keep building? We could keep going on. But the game stops at a finite number of infinite layers. It is not possible to accommodate an infinite number of infinite layers. The reason is that not all infinities are the same. I’ll cover the mathematical reasons for this in a future article. But for now, you have had a feeler for what it means to work with infinity. It is a concept that cannot be treated like just another countable number. It does not have a range (bounds), but at the same time, not all infinities are the same. If all this is confusing, look at the bright side. Every human ever known to exist can get a room at your math hotel with infinite rooms!

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Further reading that might interest you: What Really Happens When You Invent Infinite Infinities? and Why Do We Really Use The 12-Hour Clock?

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