Hilbert’s Grand Hotel

You are the desk clerk at an upscale hotel, owned by a very wealthy mathematician named David Hilbert. Hilbert’s Grand Hotel is known worldwide, and for good reason- its fame stems from its sumptuous lounge areas, excellent service, and the fact that is has infinitely many rooms.

That’s correct, the hotel you are working at has an infinite number of rooms, and thus can accommodate an infinite number of guests. I suppose there would be economic consequences to a hotel with infinite rooms, but this is not your concern as the lowly first-week desk clerk. No, what’s foremost on your mind now is the problem a new guest has just now presented to you, at the very end of your shift.

“Good evening,” she says. “I would like a room for tonight only, please.”

You gesture to the now-lit sign on the wall next to you- NO VACANCIES.

“I’m sorry, miss,” you say, “but every room is full, and you didn’t make a reservation. I’m afraid we can’t fit you in.”

The young lady, becoming aware of the increasing interest of the security staff, insists that she is a personal friend of Mr. Gilbert and that he would be very upset if she were turned away at this late hour.

Not interested in being fired due to the joint rage of this woman and Mr. Gilbert, you step away for a few moments to think of a solution.

You walk a few minutes later with a smile on your face. “Here you go, miss,” you say, handing her a key. “This room is yours. Have a good night.”

How did you do it?

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The answer hinges on the inherent weirdness of infinity and how counting works with an infinite number of things. The facts of the case are:
- Hotel has infinite number of rooms
- Rooms are “countably infinite”, that is, only whole numbers, in this case starting with 1
- All rooms are currently occupied
- You must create a vacancy for a new guest (we’ll call her Ms. K).

This is what you do. You direct every guest to pack up all their belongings and move to the next room in the hall (move from Room n to Room n+1). The hotel has infinitely many rooms, so this is possible. Then you direct Ms. K to her room (in the simple solution, Room 1) because that room is now empty.

This seems insane. The hotel has infinitely many guests already! But the hotel also has infinitely many rooms for those infinitely many guests, so it can hold any number of guests. And, by giving a general instruction instead of moving guests to specific rooms, you never have to deal with a real number and you get away with the guest-moving operation. Infinity plus one is still equal to infinity- your NO VACANCIES sign stays on.

You’ll notice that you can repeat this operation as many times as you like to accommodate any finite number of guests, much to your guests’ annoyance.

The next level is to accommodate an infinite number of guests at once, and this is where the desk clerk story falls apart a little bit- can you imagine trying to direct a hotel lobby of infinite people?- but it is possible to find vacancies for an infinite number of people at once. Instead of directing guests to Room n+1, as you did previously, you direct guests to change their room from Room n to Room 2n. Now the guest in Room 1 has moved to Room 2, Room 2 has moved to Room 4, et cetera, and now every odd-numbered room is free. And since there are a countably infinite number of odd rooms in the countably infinite set of all numbers, you’ve freed up an infinite number of rooms- simply direct your infinite new guests to go to every odd room.

These are the two simplest solutions to the Hilbert’s Hotel problem. There are also solutions that involve an infinite number of cars with an infinite number of people inside, but that one gets a little more complicated and I don’t know enough number theory or finite math to explain it away — maybe next time.

This is probably my favorite mathematical paradox, and it belongs to a category called a veridical paradox- something that initially appears false but is provably true. The thought experiment was first presented by the eponymous David Gilbert in 1924 and has been finding its way into set theory, pop culture, and paradox-lovers’ hearts ever since.