Kickstart: Ants on a stick

This is a twist of the well known ant problem, whereby we can view ants as walking through each other when they meet. In this case, we do not seek the latest time for an ant to fall off the stick, instead we want the order by which they fall off.

The first thing I noticed was that one of the leftmost or rightmost ants, $l$ and $r$ respectively, will fall off first. This is because they will either walk straight off, or they will walk into some ant and then walk towards the end of the stick. Now we just need to decide which of the two falls off first. 

Since, we can view the ants as walking past each other when meeting, we know that the $n$ ants fall off the stick at times $t(1),t(2),...,t(n)$, where $t(i)$ is the time that the $i$th ant falls off the stick when it is alone on the stick. Then the first ant will fall off at time $\min_{i}\{t(i)\}$. Let $x$ be the ant which has this minimum value of $t(x)$. If $x$ and $l$ are facing each other, then $l$ will fall off first since $x$ and $l$ will meet at some time, forcing $l$ to fall off the stick first. There are other cases to examine, but we have now made the main observations.