A short history of a long corridor

The hotel opened in 1924, when mathematician David Hilbert described it in a lecture: a hotel with infinitely many rooms, numbered without end, every one occupied.

When a single traveller arrives, the night manager shifts every guest up one room. For example, the guest in room 1 moves to room 2, 2 to 3, and so on. Room 1 is now available for the traveller.

When a coach of infinitely many passengers arrived, guests moved to double their room number. For example, the guest in room 1 moves to room 2, 2 to 4, and so on. The odd-numbered rooms (3, 5, 7, ...) are now available for the coach party!

On special occasions, infinitely many coaches, each with infinitely many passengers, have arrived. Guests moved to the powers of two, and each coach was assigned a prime number. For example, coach 1, seat 1 went to room 2^1 = 2, coach 1, seat 2 went to room 2^2 = 4, coach 2, seat 1 went to room 3^1 = 3, and so on. No two guests will have the same room.

But there is a catch. This method works only when the guests can be listed, one, two, three, and so on. When a party arrived whose members could not be listed, where the number of guests was the same as the number of points on a line, the night manager will face a challenge. Some infinities are larger than others, and a hotel, even this one, is only countable.

The story spread by George Gamow in his 1947 book One, Two, Three... Infinity. Then in 2014, a TED-Ed film by Jeff Dekofsky and David Madore brought the story to a new generation. This website is its latest guest.