Hilbert Hotel: what if countably many buses each with countably many guests arrived? Situation: There's a hotel owner David Hilbert who owns a hotel with countably many (infinity that can be mapped by natural number surjectively) rooms, and there are countable guests who lived inside starting from room NO:1 and so on.
Now if a guest arrived...I guess we all know what to do--ask everyone to move over to the next door and spare Room NO:1 for the guest.
Now if a bus of countably new guests arrived...we can ask the residents to move to rooms with room number double of their current room number so as to save all the odd-numbered room for the new guests.
Question: But now what if countably many buses each with countably many guests arrived? What can we do to find new rooms for the new guests?
I know that the union of countably many countable sets is countable, and so far I am thinking about something to do with prime number factorization raise to the power of the number of their buses...but then how do I ask the occupants to move...?
Any thoughts or better room-assigning scheme?
 A: You can have the occupants move in the same way (double their room number), then ask the new guests to take a room based on a diagonalization argument: each bus has a row in an infinite array, so the person in (1,1) takes the first open room, then (1,2), then (2,1), and so on.
A: Since academic salaries are not generous, Hilbert likes his hotel to have full occupancy.
Assume the hotel rooms are numbered $1$, $2$, $3$, and so on.
Move the guest currently occupying room $k$ to room $2k-1$. 
The $k$-th person in bus $1$ goes to room $2^1(2k-1)$.
The $k$-th person in bus $2$ goes to room $2^2(2k-1)$.
In general, the $k$-th person in bus $j$ goes to room $2^j(2k-1)$.
We have full occupancy again. 
A: Move all your guests from the $n$-th room to the $2^n$-th room.
The guests from the $k$-th bus will be placed into the powers of the $k+1$-th prime number. This ensures that all the guests are well-placed, and whatnot.
Alternatively, fix some bijection between $\mathbb N$ and $\mathbb N^2$, and send the guests to the rooms whose numbers are mapped to $(0,n)$ and send the guests from the $k$-th bus to the rooms whose numbers are mapped to $(k,n)$.
