Interesting Question on Ants
A horizontal stick is one metre long. Fifty ants are placed in random positions on the stick, pointing in random directions. The ants crawl head first along the stick, moving at one metre per minute. If an ant reaches the end of the stick, it falls off. If two ants meet, they both change direction. How long do you have to wait to be sure that all the ants have fallen off the stick?
Solution 1:
One minute at most. Imagine that ants "go through" each other. Whether the ants bounce off each other or walk past each other without changing direction has the same end effect: we have two ants approaching each other, then they meet, then they diverge from each other with that same speed of $1\text{m}/\text{min}$. So just assume that they do not bounce, but instead they keep walking the same direction, so it will take at most $1$ minute before they all fall off the edge.
Judging from the comments, it seems people disagree with the above argument, or have difficulty following/understanding it, so I offer a different interpretation. Say each ant carries a piece of paper with a written number on it. (You may call that piece of paper a baton, think of a relay race.) At the beginning all ants are numbered (from $1$ to $n$), and each ant holds a piece of paper with its own number on it. When two ants meet, they exchange their pieces of paper, and then bounce from each other. Note that while ants change direction when they bounce, the pieces of paper do not change direction. Thus these pieces of paper go with constant speed and direction $1 \text{m}/\text{min}$, so all the pieces of paper would fall off the stick in less than a minute. But, if there are no pieces of paper left on the stick, there are no ants either.
Just a comment on the first approach (when you could think ants pass by each other, instead of bouncing). Imagine that all ants look alike so much that you can't really tell them from each other. So two ants meet and bounce. But how could you tell, if you can't tell which ant is which? Perhaps you thought they bounced, but in reality each continued in its path without changing direction, passing by each other, and you do not know which is the case since these ants look so identical that you can't tell what exactly happened. Say ant A was going to the right towards ant B, and ant B was going to the left towards ant A. They meet and at the next moment you see them diverging from each other but you can't tell which ant is ant A and which ant is ant B. Perhaps the ant that now goes to the left is ant A, perhaps it is ant B. If it is ant A then they must have bounced, but if it is ant B then they must have passed next to each other without changing direction. But, it doesn't matter, since the "end result" is the same: immediately after they meet, we have two ants diverging from each other whether they bounced or they passed by each other. So assume now you have a different problem, in which ants do not bounce, but instead pass by each other (so each ant just keeps going, without changing direction). Clearly in this version of the problem all ants clear the stick in at most one minute. But, if you can't tell ants from each other, then you can't tell the two problems from each other either, so the answer to your original problem is at most a minute.
Since you put forward the name of Einstein in one of the comments, I feel entitled to involve physics in my answer. If the ants are particles, then they bounce from each other. But, if the ants are waves, then they "go through" each other, or pass next to each other. So, does light consist of particles, or waves, was Newton right (with his corpuscular theory of light), or was Huygens right (with his wave theory of light), and how is it that both theories are right?
Solution 2:
This should help arrive at the answer. Let's first consider the shortest amount of time you'd have to wait. Assuming the ants were evenly distributed, and each ant faced the nearest edge, you'd have to wait 30 seconds. This is because the farthest an ant would have to walk is 50cm.
Now this should help see what the longest path an ant would need to walk is. Assume an ant starts at Edge A and faces Edge B. If this ant walks all the way to Edge B (which takes one minute), you can be sure that there are no ants behind this ant. Now let's say this ant bumps into something right at Edge B and has to turn around. We now know that the ant will fall off Edge A after one minute. So this ant walked two minutes and we know the ruler is empty. The problem here is that when this ant gets to Edge B there's nothing to bump into because all of the ants have already fallen off at Edge B. We know this because if any were going the opposite direction, this ant would have bumped into one beforehand.
If all of the ants are facing the same direction (a 2 in 2^50 chance), you'll see one ant walk the whole length of the ruler. You know this takes one minute. The question then becomes, what's the maximum distance an ant can walk if not all ants face the same direction. In this scenario, if an ant starts at Edge A facing Edge B, this ant can only get as far as the middle of the ruler without bumping into another ant. If the ant gets past the middle, that means all of the ants in front of it didn't turn around. But we assumed at least one ant was going the opposite direction. So assume the ant does get to the middle and bumps into another ant. Because this ant started at Edge A, and made it to the middle, we know there are no ants behind it, so it has an unimpeded path back to Edge A. This total journey takes half a meter twice, or one minute.
Finally, what about the ant it bumped into at the middle of the ruler? There's no difference between that ant and the ant that started from Edge A. That ant walks for one minute.