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

[kraigandrews]

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

 

[cepheid]

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.

 

[mtayab1994]

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.