As you might expect, the Infinity Hotel, just around the corner from the Jolly Quiz Master Arms, has an infinite number of rooms.
One evening, when an extremely popular quiz night is taking place in the Jolly Quiz Master Arms, the hotel is fully booked. Unfortunately the Quiz Master at the quiz night had forgotten to make a reservation.
How did the hotel fit him in?
The hotel manager instructed all the guests to move to the room with a number one greater than their current room. So the guest in room one moved to room two, the guest in room two moved to room three, and so on with the guest in room n moving to room n+1.
As the hotel has an infinite number of rooms this can be done without a problem, and room one is now free for the Quiz Master.
This was accomplished and the Quiz Master had a room for the night, but the hotel's problems were far from over.
The quiz nights in the the Jolly Quiz Master Arms are very popular, and Infinity Quiz Tours arrived with a party of quiz enthusiasts. The bus held an infinite number of people destined for the quiz night, and all needing a room.
How did the hotel fit them all in?
This time the hotel manager instructed all the guests to move to the room with a number twice that of their current room. So the guest in room one moved to room two, the guest in room two moved to room four, and so on with the guest in room n moving to room 2n.
Then the hotel manager instructed all the passengers to go to the room with a number twice their seat number minus one. So the passenger in seat one went to room one, the passenger in seat two went to room three, and so on with the passenger in room n going to room 2n-1.
This was originally published by David Hilbert as The Paradox of the Grand Hotel