( If their previous room number was n , this time they would move to room number 2n . ) This would send the guest in room 1 to room 2 , the guest in room 2 to room 4 , the guest in room 3 to room 6 , the guest in room 4 to room 8 , and so on .
It is also possible to accommodate a countably infinite number of new guests : just move the person occupying room 1 to room 2 , the guest occupying room 2 to room 4 , and in general room n to room 2n , and all the odd-numbered rooms will be free for the new guests .