Infinite Hotel Paradox: Hilbert’s Riddle and Room 13

The infinite hotel paradox, devised by mathematician David Hilbert in 1924, demonstrates that a hotel with an infinite number of rooms can always find space for more guests—even when every room is occupied—by asking each guest to move to the next room (Stanford Encyclopedia of Philosophy). This thought experiment reveals how infinity defies our everyday logic, and it stands in curious contrast to the very finite superstition that leads many real-world hotels to skip room number 13. This article explains the paradox step by step, answers common questions, and explores why a number feared by millions of guests is quietly erased from hotel floor plans.

Last checked: 2026-08-23

Name: Hilbert’s Infinite Hotel Paradox · Concept: A thought experiment showing that a fully occupied infinite hotel can still accept more guests · Origin: Proposed by German mathematician David Hilbert in 1924 · Key trick: Move each guest to room n+1 to free up room 1

What Is the Answer to the Infinite Hotel Riddle?

The answer is that a fully occupied hotel with infinitely many rooms can always accommodate new guests—as many as countably infinite—by systematically shifting existing guests. The key is that the set of natural numbers (1, 2, 3, …) can be mapped onto a proper subset of itself, such as the set of even numbers, without changing its cardinality.

The Setup: A Hotel with Infinite Rooms

Imagine a hotel where rooms are numbered 1, 2, 3, … without end. Every room is occupied. This is a “full” hotel, but in the realm of infinite sets, “full” does not mean “no vacancy.” The Open University describes infinity in this context as “stretchable and squashable” because guests can be remapped to different rooms without changing the total number of guests.

The Solution: Making Room for One More Guest

When a new guest arrives, the manager asks every current guest to move from room n to room n+1. Guest in room 1 goes to room 2, guest in room 2 goes to room 3, and so on. Room 1 becomes free. The hotel remains fully occupied, and the new guest checks in. The Stanford Encyclopedia of Philosophy notes that this procedure works because for every natural number n there is a natural number n+1.

The Generalisation: Accommodating Infinitely Many New Guests

If a bus with infinitely many new guests arrives, the manager can free up all odd-numbered rooms by moving each current guest from room n to room 2n. This leaves the odd-numbered rooms (1, 3, 5, …) empty—countably infinite many—ready for the newcomers. The Encyclopaedia Britannica illustrates this maneuver in a short animated video. The pattern can be repeated for any finite or countably infinite number of new arrivals.

Step-by-Step: How the Infinite Hotel Paradox Works

1. Set up the infinite hotel. Rooms are numbered 1, 2, 3, … infinitely. All rooms are occupied.
2. New guest arrives. The manager instructs each guest to move from room n to room n+1.
3. Room 1 is vacated. The new guest occupies room 1. The hotel remains fully occupied.
4. To free infinitely many rooms for a busload of new guests: Move each current guest from room n to room 2n. All odd-numbered rooms become empty.
5. Repeat as needed. The same shifting logic can be applied recursively to accommodate any finite or countably infinite number of additional guests.

This step-by-step logic is the core of Hilbert’s thought experiment, as explained by ScienceABC and other popular science sources. The key insight is that the set of natural numbers has the same size as its proper subsets, a property that only infinite sets possess.

Can the Infinite Hotel Be Full?

Yes, in the mathematical sense, the hotel is “full” because every room is occupied. Yet it can still accept more guests. This apparent contradiction is resolved by understanding that “full” in an infinite set does not imply “no more room.”

What “Full” Means in an Infinite Context

In a finite hotel, a full occupancy means every room is taken and no more guests can be accommodated without evicting someone. In an infinite hotel, the condition “every room is occupied” is a bijection between the set of guests and the set of natural numbers. Wikipedia notes that the key is that the statements “every room has a guest” and “no more guests can be accommodated” are not equivalent for infinite sets.

How Hilbert’s Paradox Resolves the Contradiction

The paradox shows that the cardinality (size) of an infinite set can remain unchanged even after adding countably many elements. The University of Pittsburgh explains that Hilbert’s Hotel is a standard example of how an infinite set can be equinumerous with a proper subset of itself. The hotel is always “full” in the sense of having a bijection to ℕ, yet it can absorb new guests by reindexing.

Why Is Number 13 Not Used in Hotels?

While the infinite hotel paradox involves an infinite number of rooms, real hotels often skip a single number: 13. This practice is driven by triskaidekaphobia—the fear of the number 13—which affects a significant portion of hotel guests.

The History of Triskaidekaphobia

Wikipedia defines triskaidekaphobia as the fear or extreme superstition regarding the number 13. The term comes from Greek words for “thirteen” and “fear.” The superstition has ancient roots, including Babylonian associations with death and Norse mythology where Loki was the 13th guest at a feast, causing chaos. In Christian tradition, Judas Iscariot was the 13th guest at the Last Supper.

Hotel Practices: Skipping Room 13

Many hotels worldwide omit room number 13 and the 13th floor. A 2007 Gallup poll cited by Wikipedia found that 13% of American adults would be bothered if given a hotel room on the 13th floor, and 9% would request a different floor. News18 reports that “almost all” major hotels in many parts of the world omit room 13 or a labeled 13th floor due to guest superstition, although the floor physically exists. Some hotels label the 13th floor as 12A or M to avoid the number.

Connection to the Infinite Hotel Paradox

The infinite hotel paradox is a purely mathematical exercise about uncountable capacity, while the omission of room 13 is a commercial response to a finite, irrational fear. Both involve numbers, but one is a rigorous logical demonstration and the other is a cultural practice. The University of Pittsburgh notes that the contrast between the two reveals how humans handle abstract infinities inconsistently compared with emotionally charged finite numbers.

How Did 13 Become Unlucky?

The superstition around the number 13 has multiple historical and religious origins that have combined to make it unlucky in Western culture.

Religious Origins: The Last Supper

In Christian tradition, the Last Supper had 13 attendees: Jesus and his 12 apostles. Judas, the betrayer, was the 13th guest. This association of 13 with betrayal and doom has been a powerful influence on Western attitudes toward the number. News18 traces the superstition further back to ancient Babylonian and Norse mythology, where 13 was linked to death and chaos.

Historical Events: Friday the 13th

The combination of Friday (the day of Jesus’ crucifixion) and the number 13 creates the double superstition of Friday the 13th. The Wikipedia entry on triskaidekaphobia notes that fear of Friday the 13th is particularly strong in English-speaking countries, and some people will avoid travel, work, or major decisions on that day.

Cultural Spread and Modern Usage

The superstition has spread beyond religion into everyday life. Many buildings skip the 13th floor, airlines omit row 13, and hotels avoid room 13. A numerology-focused site notes that the practice predates Christianity and highlights Loki as a disruptive 13th figure in Norse myth who reinforces 13 as a symbol of chaos. In contrast, several East Asian cultures are more concerned with the number 4, which can sound like “death” in local languages, leading hotels there to omit floors 4, 14, and 24 (Wikipedia).

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Frequently Asked Questions

Sources cited

• Wikipedia – Hilbert’s paradox of the Grand Hotel
• Stanford Encyclopedia of Philosophy – Infinity
• Open University – Hilbert’s Infinite Hotel
• Encyclopaedia Britannica – David Hilbert’s infinite hotel paradox (video)
• ScienceABC – Hilbert’s Infinite Hotel Paradox: Explained Simply
• arXiv – The True (?) Story of Hilbert’s Infinite Hotel
• University of Pittsburgh – Infinity paradox (Hilbert Hotel)
• Wikipedia – Triskaidekaphobia
• News18 – Why Hotels Across The World Omit Rooms With Number 13
• Esoteric Numbers – Hotel Room Superstition Explained by Numerology
• Wiris – Hilbert’s Paradox Explained: The Infinite Hotel Mystery