Variant 1
The well-renowned hotel of Bilzebab is famous for having an infinite number of rooms, and, being such a famous and well-renowned hotel, the hotel is occupied.
Ten tourists arrive at the hotel and ask if there are any rooms available for them.
Bilzebab is also famous for having a smart and quick-thinking manager. The manager thinks for a minute, and then says:
“Yes, of course, I’ll just reorganize the current arrangements, and you will all get a room”.
In which rooms will the newcomers stay?
Please note: After the manager has reorganized the room arrangements, no one will share a room, no one will be thrown out, and the hotel will still be fully occupied.
Each person, including the one already with rooms, will have to be given a specific, calculable, new room number. Saying “go to room infinity plus 1” is not valid.
Answer
The hotel manager can ask everyone to move 10 rooms up - the person staying in room 1 will go to room 11, the one in room 2 will go to 12, and so on. This leaves rooms 1 to 10 available for the newcomers.
Variant 2
Same setup as for Variant 1, but an infinite number of tourists arrive at the hotel and ask if there are any rooms available for them.
In which rooms will the newcomers stay?
Answer
The hotel manager can ask everyone to take their room number, double it, and move to that room. The person in room 1 will go to room 2, 2 will go to 4, 3 to 6, and so on. This leaves all the odd-numbered-rooms available - an infinite number of rooms. The first newcomer can go to the first odd-numbered room (room 1), the second can got to the second (room 3), and so on.
Variant 3
Same setup as for Variant 1, but an infinite number of buses, each with an infinite number of tourists, arrive at the hotel and ask if there are any rooms available for them.
In which rooms will the newcomers stay?
Answer
The solution to this is best described by picturing a matrix, with an infinite number of columns and rows. Each row is a bus, and each cell on a row is a person in that bus. We can consider the people already in the hotel as a zeroth bus - row one in the matrix.
We can now go in diagonals over the matrix, north-east to south-west, starting in the upper left corner. Person (1,1) will get room 1, person (1,2) will get room two, person (2,1) room three, and so on.