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)

Comments

  1. Gold and Iron2:07 PM, December 16, 2009

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

    ????

    ReplyDelete
  2. Pratik Poddar12:29 AM, December 18, 2009

    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.

    ReplyDelete
  3. The Sherminator1:24 PM, October 11, 2010

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

    ReplyDelete
  4. Pratik Poddar2:31 PM, October 11, 2010

    @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:

    http://en.wikipedia.org/wiki/Cantor_pairing_function

    Thanks!

    ReplyDelete
  5. Unknown5:07 PM, October 02, 2014

    @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...

    ReplyDelete
  6. Unknown5:10 PM, October 02, 2014

    Infinite Cartesian product of countable sets is uncountable

    ReplyDelete
  7. Unknown5:23 PM, October 02, 2014

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

    ReplyDelete
  8. Pratik Poddar1:05 PM, September 08, 2026

    Not quite - infinitely many buses each with infinitely many people is N x N, not N^N. It is a countable union of countable sets (equivalently, pairs of natural numbers), which is still countable - the Cantor pairing function maps it back to N, and that is why the hotel can take them all. (replied using AI)

    ReplyDelete
  9. Pratik Poddar1:05 PM, September 08, 2026

    True - an infinite Cartesian product of countable sets (like N^N) is uncountable. But that is not what infinitely many buses of infinitely many people gives you: that is a countable UNION of countable sets, N x N, which is countable. The distinction between product and union is the whole game here. (replied using AI)

    ReplyDelete
  10. Pratik Poddar1:06 PM, September 08, 2026

    Exactly. Countable union of countable sets is countable - so Hilbert's hotel absorbs infinitely many buses of infinitely many guests with room to spare. (replied using AI)

    ReplyDelete

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