room 01728394 (i.e., room 1,728,394). Unlike the prime powers solution, this one fills the hotel completely, and we can reconstruct a guest's original coach and seat by reversing the interleaving process. First add a leading zero if the room has an odd number of digits. Then de-interleave the number into two numbers: the coach number consists of the odd-numbered digits and the seat number is the even-numbered ones. Of course, the original encoding is arbitrary, and the roles of the two numbers can be reversed (seat-odd and coach-even), so long as it is applied consistently. Triangular number method Those already in the hotel will be moved to room , or the th triangular number. Those in a coach will be in room , or the triangular number plus . In this way all the rooms will be filled by one, and only one, guest. This pairing function can be demonstrated visually by structuring the hotel as a one-room-deep, infinitely tall pyramid. The pyramid's topmost row is a single room: room 1; its second row is rooms 2 and 3; and so on. The column formed by the set of rightmost rooms will correspond to the triangular numbers. Once they are filled (by the hotel's redistributed occupants), the remaining empty rooms form the shape of a pyramid exactly identical to the original shape. Thus, the process can be repeated for each infinite set. Doing this one at a time for each coach would require an infinite number of steps, but by using the prior formulas, a guest can determine what their room "will be" once their coach has been reached in the process, and can simply go there immediately. Arbitrary enumeration method Let . is countable since is countable, hence we may enumerate its elements . Now if , assign the th guest of the th coach to the th room (consider the guests already in the hotel as guests of the th coach). Thus we have a function assigning each person to a room; furthermore, this assignment does not skip over any rooms. Further layers of infinity Suppose the hotel is next to an ocean, and an infinite number of car ferries arrive, each bearing an infinite number of coaches, each with an infinite number of passengers. This is a situation involving three "levels" of infinity, and it can be solved by extensions of any of the previous solutions. The