I encountered this puzzle many years ago, and I think back on it often as it is unique and thought provoking. As far as I know nobody has proved the given solution as optimum, so it may still be improved upon. I will provide the link to the original problem at the end.
I did reduce the complexity of the problem so that I can be sure of the solution to my less-intense version. Here's my version:
A hotel developer is building a new hotel, but he doesn't like elevators very much - he gets very confused by all the buttons, so he decided all of his elevators will only have one button. Long story short, each elevator will only stop on two floors. Being a sensible man, he will build more than one elevator shaft - six of them to be precise. He will also be using new MagLev elevator transport system - that means no cables and that means more than one elevator can be housed in a single shaft, as long as they don't have intersecting domains (he doesn't want any collisions!). For example: in the first shaft an elevator can be built that goes from floor 1 to floor 4. Only those two floors are serviced by that elevator. However in the same shaft a second elevator can be housed that will go from floor 5 to floor 7, and only those two floors, etc. The developer wants to be able to get from any floor in his hotel to any other floor with no more than 3 elevator trips. So, again, as an example: a guest can take elevator in the first shaft from floor 1 to floor 4, then take an elevator in the second shaft from floor 4 to floor 5 and then take the elevator in the first shaft again from floor 5 to floor 7. That will be 3 trips and the end of his adventure. Of course all the floors in the hotel need to be serviced! The question is what is the maximum number of floors that this developer can build for this hotel with these conditions?
If my version is a little too easy for you, try the harder version:
Same conditions as before, except each elevator can now stop at 6 floors.
Here's the link to where I first saw this: Elevator Efficiency
A solution has been found to the hard problem, but, like I said, nobody has proved that it is optimal, so maybe there's an obscure trick that can increase the floor number. As for the 2 floor problem I'll leave a hint:
Dr. Xorile had the right idea of making a hub and spoke system, but rot13(Lbh pna unir zber guna 3 uhof. Gur gevpx vf gung zber guna bar uho pna or npprffvoyr sebz pregnva sybbef.)