Another flavour of Hilbert's Hotel Many of you have probably heard about Hilbert's Hotel problem. Mr Hilbert owns a hotel with countably infinite amount of one-bed rooms. All the rooms are, of course, taken.

A (finite or infinite) group of k people walks in and wishes for accommodation. However, here comes the tricky part. The current guests are quite tired and Mr Hilbert does not wish to make them move from one room to another. How does he achieve accommodating the new batch of people without moving the already accommodated ones?

This problem, although it's quite known and documented is nowhere to be found here, or anywhere else on the Internet (which is kind of strange).
My understanding of the Hilbert Hotel is that the hotel has $n$ rooms. Since the number of rooms is infinite a new guest will get the room $n_{+1}$. So if $k$ people arrive the will stay in the rooms $n_{+1},...,n_{+k}$, and there is no need for the other guests to move.
The usual specification is that the hotel is already full when the new guests arrive.  If that is true, you cannot accommodate them without moving the guests that are already there.  If we just have to have an infinite number of guests in the hotel and accommodate a number more without moving the existing ones, just put the existing guests in the odd numbered rooms and put the new guests in the even numbered rooms.  The question is not clearly stated.