Hilbert's Hotel Calculator
Hilbert's Hotel Calculator illustrates how a fully booked hotel with infinitely many rooms can still fit more guests by shifting everyone along. Enter how many new guests arrive.
Formula
Shift each guest from room n to room n + k, freeing rooms 1…k
- A hotel with infinitely many rooms is never truly full.
- Moving guest n to room n+k opens k rooms while keeping everyone housed.
5 new guests
Inputs
- New Guests Arriving: 5
- A Current Guest in Room: 1
The guest in room 1 moves to room 6; rooms 1–5 open up for the new arrivals.
Frequently asked questions
What is Hilbert's Hotel?
A thought experiment showing a full hotel with infinitely many rooms can still accept new guests by shifting everyone to higher-numbered rooms.