You are on the bottom floor (floor 0, lets call it) of an apartment building with no basement. There are n elevators, which we index 1,2,…,n. Assume the elevators are on floors f1, f2, … , where fk > 0 is the floor elevator k is currently on, 1 <= k <= n. Assume you only like one of the elevators, elevator e.
The way the elevator logic works is this: When you press the elevator button, one closest to you ( = one on floor number min(fk, k>0)) is told to go to 0. If there is a tie then, of those on the same lowest floor, the elevator with the smallest index is told to go to 0.
Move: If you press the button and some other elevator than elevator e arrives, you can tell it to go to any floor you wish.
Taboo: You can press the elevator button if and only if no elevator is moving down.
Goal: You want to use elevator e (for some fixed e=1, 2, …, n).
Problem: Is there a finite sequence of moves that allows you to ride in elevator e?
My plan is to post the answer sometime later, but have fun with it!
Yet Another Mathblog 
Here’s a solution and it’s mostly good news! You can get your favorite elevator e to come down… unless it’s already sitting at the very top floor M. If elevator e starts at the top, you’re out of luck unless e=1.Why? Because being at the top means being as high as possible, but the system chooses the lowest elevator. The only way elevator e at the top could be chosen is if every elevator werealso at the top—but then the tiebreaker would pick the elevator with the smallest number, whichisn’t e (unless e = 1).The Winning Strategy (assuming elevator e is not at the top floor): Press the button. Some elevator i will come down. If i=e then you’re done, but if i not= e, then press the button and send it up to height M. Each time you execute this step you’re removing one “problematic” elevator from the competition by parking it on the top floor M.Since there are only n – 1 other elevators (besides your favorite e), you can do this at most n–1 times. Eventually, all the other elevators will be sitting at the top floor. Press the button one last time to get your favorite one.Pretty neat for something that started with just pressing an elevator button, right?