Infinity Problem

Suppose you have a hotel which has one floor with infinite number of rooms in a row and all of them are occupied. A new customer wants to check in, how will you accommodate her? What if infinite number of people want to check in, how will you accommodate them then?

Hint: Define infinity. :)

Update (18/12/09): Solution in comments by Nikhil Pandey (CSE, IITB alumnus)


  1. 1) ask everyone to move to next room
    2) ask everyone to move from x->2x


  2. pandeyji.. (@Gold and Iron)
    sahi jawaab.. dhanyawaad. :)

    Since there are infinite number of rooms and infinite+1= infinite
    Just ask person in room k to move to k+1. Thus making a room in the first room. :)

    In the other case, since infinite+infinite = infinite
    asking person in room k to move to 2k solves the problem.

  3. Third part: Suppose infinite number of buses arrive at the hotel, each having infinite number of people, how will you accommodate them? :)

  4. @Sherminator..
    Nice one..
    Since NxN is countable set. We can get a one to one mapping from N to NxN
    Hence, we can accommodate (infinite people X infinite buses) in the hotel.

    Relevant article:


  5. @Pratik . Infinite number of buses having infinite number of people would make the set N^N i.e. N x N x N..... N times...

  6. Infinite Cartesian product of countable sets is uncountable

  7. Oh got it . Its a countable union of countable set.


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