David Hilbert’s thought experiment shows that infinite sets behave strangely: a full hotel with infinitely many rooms can still take in new guests – even infinitely many busloads – just by moving everyone around.
How to use the Hilbert’s Hotel Paradox Calculator
- Scenario
- Current guest’s room n
- Number of new guests k
- Bus number b
- Seat number s of the new guest
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Choose the scenario and enter the current guest’s room (and the bus and seat for new arrivals). The calculator gives the new room numbers.
Formula
k new guests: n → n + k · one infinite bus: n → 2n, passenger s → 2s − 1
Infinitely many buses: guest n → 2ⁿ, passenger s of bus b → p_bˢ (p_b the (b + 1)-th prime)
Worked example
Inputs — Scenario: A finite number of new guests; Current guest's room n: 3; Number of new guests k: 5.
Results — New room of the current guest = 8; Room of the new guest(s) = Rooms 1 to 5; How it works = Every guest moves from room n to room n + 5, freeing the first 5 rooms..
Key terms
- Countable infinity
- The size of the set of natural numbers.
Frequently asked questions
Is the hotel ever full?
It is always "full", yet it can always take more guests.
Why does the prime-power method work?
Unique factorisation means no two guests get the same room.
Are all infinities the same size?
No – the real numbers are a bigger infinity; Hilbert's hotel holds only countable ones.
Who was Hilbert?
David Hilbert, one of the most influential mathematicians of the early 20th century.