Created by Gabriela Diaz
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: Jun 26, 2023

Welcome to Hilbert's hotel paradox calculator! This hotel is always full, but it can still accommodate an infinite number of new guests. Are you wondering how this is possible? Well, the mathematician David Hilbert pondered this same question and came up with a fascinating solution. Using our calculator, you can explore different scenarios and discover how this infinite hotel works 🛎

• What the infinite hotel paradox is;

• How to accommodate infinitely many new guests;

• How to use this Hilbert's grand hotel calculator; and

• What the transfinite numbers are.

So let's dive into the mysteries of infinity!

🙋 In this calculator, we use the term "infinitely many" to refer to countably infinitely many. In this paradox, the set $S$ (which corresponds to the infinite number of hotel rooms) is of the same size as the set of natural numbers $\N = \{1, 2, 3, ...\}$.

## What's the infinite hotel paradox?

The infinite hotel paradox, also known as Hilbert's hotel paradox, is a thought experiment that explores the fascinating concept of infinity.

Imagine a hotel with infinitely many rooms numbered 1, 2, 3, and so on. Now, suppose the hotel is completely full, with a guest occupying every room. It seems logical that there's no room available for a new guest. Well, prepare to be amazed!

In this infinity paradox, even though the hotel is full, it can still accommodate new guests. How is this possible, you may wonder? Since the hotel's capacity is infinite, there is no final room number. This means that the hotel manager can accommodate not only a few but any number of new guests — be it 1, 2, 3, or even an infinite number.

🙋 Are you enjoying this infinity paradox? We have more to explore! Check Galileo's paradox of infinity calculator for additional infinite fun!

## How to accommodate the new guests?

So far, we know that the hotel is always "full," and that somehow, the manager still manages to accommodate a finite or infinite number of new guests. But how can the manager achieve this seemingly impossible task? The answer depends on the details of the scenario — let's explore some!

#### Finitely many new guests

Let's start with a scenario involving a finite number of new guests. In this case, the manager simply shifts every current guest to a room with a higher number. The number of rooms every current guest is shifted is equal to the number of new guests. By doing this, we empty enough rooms at the lower end of the room number range, so that the new guests may occupy them. We can express this using the following equation:

\scriptsize \begin{aligned} \text{Cur}&\text{rrent guest's new room} =\\ &\text{Current guest's current room} \\ +\ \ & \text{Number of new guests} \end{aligned}

For example, if two new guests enter the hotel, all existing guests will move up two room numbers, and the new guests will occupy rooms 1 and 2.

#### Infinite new guests

But what happens when a bus arrives with an infinite number of new guests? The previous solution won't work:

• The manager can't assign specific room numbers to each current guest.
• How many rooms should they even move, since there are infinitely many new guests?

So, the manager comes up with a clever plan and instructs every current guest to move to an even-numbered room that is double their current room number:

$\scriptsize \begin{split} \textrm{Current} & \textrm{ guest} \\[-3pt] \textrm{new} & \textrm{ room } \end{split} = 2 \times \begin{split} \textrm{Current} & \textrm{ guest} \\[-3pt] \textrm{current} & \textrm{ room} \end{split}$

As for the new guests, they are assigned to the empty odd-numbered rooms based on their seat numbers on the bus:

$\scriptsize \begin{split} \textrm{New} & \textrm{ guest} \\[-3pt] \textrm{room } & \textrm{number} \end{split} = \left( {2 \times \begin{split} \textrm{Se} & \textrm{at} \\[-3pt] \textrm{nu} & \textrm{mber} \end{split}} \right) - 1$

#### Infinitely many buses with infinitely many guests

Now, let's imagine a more challenging situation: not just one bus arrives, but an infinite number of buses with an infinite number of passengers each. How can the manager handle this? Fortunately, our clever manager has a solution. He decides to send the current guests to the rooms that correspond to the powers of 2, raised to their current room number:

$\begin{split} \textrm{\small Current guest} \\[-5pt] \textrm{ \small new room} \end{split} = 2 ^ {\left(\begin{split} \textrm{\scriptsize Current guest} \\[-3pt] \textrm{ \scriptsize current room} \end{split}\right)}$

As for the new guests, the manager assigns a prime number to each bus, and the room number for each new guest is calculated by raising the prime number corresponding to this guest's bus to the power of their seat number on the bus:

$\footnotesize \begin{split} \textrm{New} & \textrm{ guest} \\[-3pt] \textrm{room } & \textrm{number} \end{split} = \textrm{Bus prime} ^ {\scriptsize \left( \begin{split} \textrm{Se}&\textrm{at} \\ \textrm{num}&\textrm{ber} \end{split} \right)}$

By using prime numbers, the manager ensures that the room numbers won't repeat, as prime numbers are not multiples of each other. You'll note that the current guests occupy all the powers associated with the first prime number, 2. This means that the next guests will start occupying rooms related to the following prime number, which is 3.

This last solution is known as the prime power method. Still, other methods, such as the prime factorization, interleaving, or triangular number, can also help the manager solve this last problem of infinite layers or nesting infinities.

## Does infinity exist?

Infinity exists as a mathematical concept, representing the idea that something is limitless and boundless. However, we have yet to be able to prove its existence as a physical entity.

Nonetheless, infinity plays a crucial role in scientific and philosophical discussions, particularly in fields like cosmology, where we contemplate the vast expanse of the universe and consider whether space and time are finite or infinite.

🙋 One thing we know for sure is that the universe is constantly expanding. Check out our universe expansion calculator to learn more about this fascinating topic 🌌

## Using Hilbert's hotel paradox calculator

Whether you're the clever manager of an infinite hotel or simply curious about how to accommodate infinitely many guests, Hilbert's hotel calculator can help you find the perfect solution. Here's how to use this tool:

To get started, select the scenario you want to evaluate from the drop-down menu. Then, follow the steps below based on your selection:

#### Finitely many new guests

1. Enter the number of new guests and the room number of the current guest you want to move.
2. The calculator will indicate the new room number for the current guest.

#### Infinitely many new guests

1. Indicate the room number of the current guest.
2. The calculator will indicate the new room number for this guest.
3. Enter the seat number of the new guest.
4. The calculator will show the new room number for this new guest.

#### Infinitely many buses with infinitely many guests

1. Input the room number of the current guest.
2. The calculator will display the new room number for this guest.
3. Enter the bus number and seat number of the new guest.
4. The calculator will show the new room number for this new guest.

## FAQ

### Can Hilbert's hotel run out of rooms?

No, Hilbert's infinite hotel cannot run out of rooms. It doesn't matter how many guests are already accommodated — the infinite nature of the hotel ensures that there will always be an infinite number of rooms available to accommodate additional guests.

Even when the hotel is already hosting an endless number of guests and is considered "full", it can still accommodate an infinite number of new incoming guests.

### What are transfinite numbers?

Transfinite numbers are a mathematical concept where certain numbers are considered infinitely larger than finite numbers but are not necessarily absolutely infinite. This means there can be other numbers still infinitely larger than the initial "infinite" number. The concept of transfinite numbers includes transfinite cardinals and transfinite ordinals and was introduced by Georg Cantor.

### Where should the current guest at room 1397 move if infinite guests arrive?

The current guest should move to room 2794. To determine the new room number:

1. Identify the guest number you want to re-accommodate.

2. Double its current room number:

new room number = 2 × original room number

3. The result will be the new room number for the current guest.

Gabriela Diaz
At Hilbert's Infinite Hotel, new guests never stop arriving, and the rooms are always full!

As the hotel manager, your job is to find a place for every new guest. Luckily, you can easily determine where to accommodate them all with this handy tool.

Simply choose the scenario that best fits your situation and discover where to allocate your guests! 🛎
Scenario
Finitely many new guests
Finitely many new guests
How many new guests are there?
Current guest's original room number
Current guest's new room number
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