How Can All Guests Stay at Hilbert's Hotel When Even Rooms Close?

  • Thread starter kraigandrews
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In summary: Now if you were to do this same thing again, but this time the even numbered rooms are already occupied, then you can do the same thing, but instead you have to move the guest to the 2 times his room, PLUS 1. This way you will always have an odd numbered room available to move the guest to even if all the even numbered rooms are occupied. In summary, by using the same trick as before, but adding 1 to each room number, you can free up enough rooms to accommodate all the guests at Hilbert's Grand Hotel.
  • #1
kraigandrews
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Homework Statement


Suppose that Hilbert's Grand HOtel is fully occupied on the day that the hotel closes all the even numbered rooms for maintenance. Show that all guests can remain at the hotel


Homework Equations




The Attempt at a Solution


I am not quite sure how to solve this, my first thought was to move every guest in room 2n for n=1,2,3,..., to some multiple of 2n+1, but obviously these room are already occupied. I am really stumped on this one, any help is appreciated.

Thank you
 
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  • #2
I'm not an expert in Number Theory, but it seems to me that you can apply a variation on the same trick that is used to add more guests to a fully-occupied Hilbert Hotel. The standard trick is that if the hotel is fully occupied, you move all the guests to even-numbered rooms 2n (of which there are infinitely-many), thus freeing up the odd-numbered ones 2n+1 to accommodate more guests.

In this case, the even numbered rooms are no longer available, and all the odd ones are occupied. But if you think about it, this is really just the *same* situation as before. You have infinitely-many occupied rooms numbered 2n+1, and you can just move all the guests in those rooms to the ones for which n is even (i.e. 1, 5, 9, 13,...), thus freeing up all the rooms for which n is odd (3, 7, 11, 15, ...). This is equivalent to taking all the available rooms (the ones that were originally odd-numbered) and just RE-numbering them from 0 to infinity, and then choosing all the even-numbered ones in the new numbering system to move the guests to. I hope that makes sense.
 
  • #3
Alright look at it in a different way. Imagine that the hotel is fully occupied (infinity) and a bus comes along and brings (infinity amount of guests.). The way you would free up the rooms to make room for everyone is that you would make the person move to the room that is 2 times the number of his room. This way you would have freed up enough room to accommodate the guests.
 
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FAQ: How Can All Guests Stay at Hilbert's Hotel When Even Rooms Close?

What is Hilbert's Grand Hotel Paradox?

Hilbert's Grand Hotel Paradox is a thought experiment created by mathematician David Hilbert to illustrate the counterintuitive nature of infinity. It involves a hotel with an infinite number of rooms and an infinite number of guests.

How does the paradox work?

The paradox begins with a fully occupied hotel, with each guest assigned to a specific room. Then, a new guest arrives and requests a room. The hotel manager can accommodate this request by simply having each guest move to the next room number (i.e. the guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on), leaving room 1 available for the new guest. This process can be repeated infinitely to accommodate an infinite number of new guests, even though the hotel was already fully occupied.

Does this mean that infinity can exist in the real world?

No, the paradox is simply a thought experiment and does not reflect the nature of infinity in the real world. In mathematics, infinity is often used as a concept or symbol, but it does not have a tangible existence in the physical world.

What is the significance of Hilbert's Grand Hotel Paradox?

The paradox highlights the strange and counterintuitive properties of infinity, such as the fact that adding or subtracting from an infinite set does not necessarily change its size. It also raises questions about the nature of space and whether it can be infinitely divided.

Are there any real-life applications of this paradox?

Hilbert's Grand Hotel Paradox is primarily a thought experiment and does not have any direct practical applications. However, the concept of infinity is used in various fields such as mathematics, physics, and computer science, and this paradox can help illustrate some of its properties and implications.

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