# Hilbert's Hotel

Hilbert's paradox of the Grand Hotel is a mathematical thought exercise in which is imagined a hotel with infinite rooms, thus capable of accommodating infinite guests. Even if the hotel is full to capacity, and an extra guest arrives, he or she can still be accommodated by putting them into room 1408, which nobody ever stays in because the television doesn't work and the bed smells a bit funny. Problems arise because only a finite number of the rooms have ocean view, and these rooms are usually filled by mathematicians, for whom the hotel is a popular conference venue. The hotel is also known for its annual Don Con, a popular convention where infinitely many Cambridge University dons visit the hotel for a week. These dons normally live in a (finite) cupboard in the university, which is possible because they stand on each other's shoulders, and almost all of them are rather small, each being half the height of the previous don.

## Example problems involving Hilbert's Hotel[edit]

Q: The hotel is full to capacity, and an infinite number of further tourists arrive. How does the hotel accommodate them all?

A: The staff directs them to the Infinite Ritz-Carlton across town.

Q: The hotel is full to capacity. The guests in even-numbered suites (of which there are infinite) all call room service to request a VCR. How long will it take the hotel staff to oblige?

A: The hotel no longer carries VCRs, as they are obsolete technology.

Q: Donald Trump has bought Hilbert's Hotel and wishes to gold plate it. How much gold will he require?

A: Four hundred carats.

Q: An infinite number of customers refuse to pay their bills. Does the Hotel move into bankruptcy?

A: No. The IRS only have an finite number of tax accountances.

Q: Mr Trump decides to expand his empire to a chain consisting of an infinite number of Hilbert Hotels. Does this infinite chain contain more rooms than the original infinite Hotel?

A: No. He contracted a Polish construction firm.

Q: Where do the hotel staff find enough water to fill the infinite swimming pool?

A: A Klein bottle.

Q: All guests switch to the room with the number double their previous room number. Why do they do this?

A: They are so drunk that they count double.

Q: Infinite guests are staying at the hotel. When they leave, they each steal a towel, making a total of infinite towels stolen. How does the management replace the stolen towels?

A: They cut all the remaining towels in half.

Q: A fire breaks out in room 991, threatening to spread to adjacent rooms. What is the management's response when asked to call the fire brigade?

A: If you find yourself being horribly burnt, please move to double your room number, plus three, times the square root of pi. Moving rooms solves all problems.

Q: As the hotel covers an infinitely large area, its rent is infinitely high. How does the hotel recoup this cost?

A: Minibars.

Q: At the Don Con, the dons want to play a party game with uncountably many rules. Can they do this?

A: Only by assuming the axiom of choice, which is obviously rubbish.

Q: Someone asks how an infinite hotel could possibly exist given the finite size of the universe. What do the management do?

A: Shoot them on sight.

**Glossary of mathematical terms**