What is Hilbert's hotel, and how can a full hotel take more guests?

Hilbert's hotel, also called the Infinite Hotel Paradox or Hilbert's paradox of the Grand Hotel, is a thought experiment about a hotel with infinitely many rooms numbered 1, 2, 3 and so on, with no upper limit. Every room is occupied, yet the hotel can still make space for new arrivals. It is used to show how infinite collections behave differently from finite ones. [1] [2]

According to Wikipedia, the idea was introduced by the German mathematician David Hilbert in a 1924-1925 lecture titled "Über das Unendliche". It was later popularized by George Gamow's 1947 book One Two Three... Infinity. Plus Maths describes Hilbert as using a hotel to demonstrate the counter-intuitive games one can play with infinity. [1] [2]

For one new guest, the manager asks every guest to move from room n to room n+1. Because there is no last room, nobody is left without a room to move to, and room 1 becomes free. The same method handles any finite number k of new guests: each guest moves from room n to room n+k. The moves must happen simultaneously, since moving one guest after another would take an infinite amount of time. [1] [2]

For infinitely many new guests, the manager moves each guest from room n to room 2n. All the even-numbered rooms are then occupied and all the odd-numbered rooms, of which there are infinitely many, are free for the newcomers. [2] [3]

Plus Maths extends the puzzle to infinitely many coaches, each carrying infinitely many passengers. First the existing guests move to double their room numbers, freeing the odd rooms. Then the passenger in seat n of coach m is sent to room p to the power n, where p is the (m+1)st prime number. Coach 1 uses powers of 3, coach 2 uses powers of 5, coach 3 uses powers of 7, and so on. The fundamental theorem of arithmetic, which says every whole number factors into primes in only one way, guarantees that no two passengers are given the same room. [2]

The same source says guests arriving in any finite number of layers of infinity can be accommodated, but not always guests arriving in infinitely many layers of infinity. [2]

Wikipedia calls it a veridical paradox: the result seems wrong but is provably true. The statements "every room has a guest" and "no more guests can be accommodated" are equivalent for a finite hotel but not for an infinite one. The surprise comes from the fact that infinite collections have properties quite different from those of finite collections. [1]