1. Subscribertalzamir
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    01 Oct '11 19:57
    A hotel has an infinite number of rooms, which are all fully booked, but that's no problem as the guests are flexible. They are accustomed to being moved to another room, up to once a day, to make more space, as long as they are clearly told where to go.

    On the first day, a thousand new people show up and need rooms. This is easy for the manager, everyone is given a new room by moving from room n to room n + 1,000, which vacates rooms 1 .. 1,000 for the new arrivals.

    On the second day, an infinite number of new people arrive. How are they given rooms?

    On the third day, an infinite number of buses arrive, each with an infinite number of people. How does everyone now get a room?
  2. Joined
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    02 Oct '11 13:26
    Originally posted by talzamir
    A hotel has an infinite number of rooms, which are all fully booked, but that's no problem as the guests are flexible. They are accustomed to being moved to another room, up to once a day, to make more space, as long as they are clearly told where to go.

    On the first day, a thousand new people show up and need rooms. This is easy for the manager, everyon ...[text shortened]... mber of buses arrive, each with an infinite number of people. How does everyone now get a room?
    No idea... ask Hilbert, I'm sure he knows.

    Reveal Hidden Content
    Alternatingly odd and even, and in a plane-walking pattern, respectively.


    Richard
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    02 Oct '11 13:582 edits
    Reveal Hidden Content
    Number the rooms a1; a2, b1; a3, b2, c1; a4, b3, c2, d1 etc

    Reveal Hidden Content
    Guests move to "a" rooms, first bus to "b" rooms etc.


    The power is in the etc.s
  4. Standard memberforkedknight
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    03 Oct '11 05:21
    Silly problem, an infinite number of rooms can't possibly be fully booked.

    But nor can an infinite number of guests possibly be accommodated.
  5. Standard memberwolfgang59
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    03 Oct '11 07:17
    Originally posted by forkedknight
    ... an infinite number of guests possibly be accommodated.
    some guests will have to share ....
  6. Standard memberwolfgang59
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    03 Oct '11 07:17
    Originally posted by forkedknight
    Silly problem, an infinite number of rooms can't possibly be fully booked.

    .
    Some guests will have more than one room.
  7. Subscribertalzamir
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    03 Oct '11 07:50
    If infinity and hotels together feel silly, this could of course be mathified, e.g.

    Is it possible to pair up with positive integers...
    a. all natural numbers?
    b. all integers?
    c. all rational numbers?
    d. all real numbers?

    where cases a..c are just about the same as the cases of adding a finite number of new guests, an infinite number of new quests, and infinity squared number of new guests. Correct answers given already by readers on those cases.
  8. Standard memberforkedknight
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    03 Oct '11 14:16
    Originally posted by talzamir
    If infinity and hotels together feel silly, this could of course be mathified, e.g.

    Is it possible to pair up with positive integers...
    a. all natural numbers?
    b. all integers?
    c. all rational numbers?
    d. all real numbers?

    where cases a..c are just about the same as the cases of adding a finite number of new guests, an infinite number of new ques ...[text shortened]... infinity squared number of new guests. Correct answers given already by readers on those cases.
    a) yes Reveal Hidden Content
    1:1, 2:2, etc

    b) yes Reveal Hidden Content
    1:0, 2:1, 3:-1, 4:2, 5:-2, etc

    c) yes Reveal Hidden Content
    A bit more complicated, but you can walk a table diagonally 1:1/1, 2:1/2, 3:2/1, 4:1/3, 5:3/1, 6:4/1, 7:2/3, 8:3/2 etc

    d) no

    Therefore, case a,b,c are all adding the same number of guests 🙂
  9. Standard memberwolfgang59
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    04 Oct '11 02:271 edit
    Originally posted by talzamir
    A hotel has an infinite number of rooms, which are all fully booked, but that's no problem as the guests are flexible. They are accustomed to being moved to another room, up to once a day, to make more space, as long as they are clearly told where to go.

    On the first day, a thousand new people show up and need rooms. This is easy for the manager, everyon ...[text shortened]... mber of buses arrive, each with an infinite number of people. How does everyone now get a room?
    On the third day the buses are numbered as they arrive and the occupants go to the floor indicated by their bus number.

    Each floor has an infinite number of rooms to accomodate the infinite number of passengers on each bus.

    EDIT: The current guests all remain on the ground floor.
  10. Standard memberPalynka
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    04 Oct '11 12:40
    Originally posted by forkedknight
    Silly problem, an infinite number of rooms can't possibly be fully booked.

    But nor can an infinite number of guests possibly be accommodated.
    It's a countably infinite number, so bijection says otherwise.

    And...here...we...go!

    You put 2 numbered balls in a bag and...
  11. Standard memberforkedknight
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    04 Oct '11 18:001 edit
    Originally posted by Palynka
    It's a countably infinite number, so bijection says otherwise.

    And...here...we...go!

    You put 2 numbered balls in a bag and...
    Just because you take take infinitely many guests and map a room to each guest doesn't mean you can fill a hotel with infinitely many rooms. There will always be more rooms. That's the definition of infinite.
  12. Standard memberPalynka
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    04 Oct '11 23:41
    Originally posted by forkedknight
    Just because you take take infinitely many guests and map a room to each guest doesn't mean you can fill a hotel with infinitely many rooms. There will always be more rooms. That's the definition of infinite.
    If there is a bijection, then there are the same amount of rooms as guests. By definition.
  13. DonationAnthem
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    05 Oct '11 00:431 edit
    Originally posted by Palynka
    If there is a bijection, then there are the same amount of rooms as guests. By definition.
    What mathematician ever defined "same amount" via the existence of a bijection? That sounds more like the definition of cardinality to me.
  14. Standard memberforkedknight
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    05 Oct '11 02:17
    Originally posted by Palynka
    If there is a bijection, then there are the same amount of rooms as guests. By definition.
    And yet you can add 1000 more guests, and they all still have rooms.

    Using the concept of infinity as though it is a number doesn't make any sense.
  15. Standard memberPalynka
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    05 Oct '11 07:282 edits
    Originally posted by forkedknight
    And yet you can add 1000 more guests, and they all still have rooms.

    Using the concept of infinity as though it is a number doesn't make any sense.
    A finite number of elements can always be added, but you need to change the mapping else there would be no available rooms.

    What doesn't make any sense is your blatant disregard for set theory and cardinality. If two sets are linked by a bijection, they have the same cardinality, and sets with the same cardinality have the same number of elements.
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