## Problem #031 - piped ants

A bunch of ants are left inside a very, very, tight tube, and they keep colliding with each other and turning around. How long will it take them to escape?

# Problem statement

Suppose you have a very thin pipe. The pipe is the exact width of an ant, so that ants can travel inside the pipe, but they can't cross paths because the pipe isn't wide enough for that. Assume an ant takes exactly 1h to travel through said pipe.

I will now place several ants along the pipe, facing different directions, and all ants will start walking in a straight line. When two ants collide, they both turn around and continue walking in the opposite direction they were walking in. When an ant reaches the end of the pipe, the ant just leaves the pipe.

What is the shortest amount of time that ensures all ants have left the pipe?

# Solution

This problem may look quite confusing if you try to imagine all the interactions/bumps there might be between an arbitrary number of ants inside the pipe. However, the only thing that matters to us is the time each ant takes to leave the pipe, and not the number of times the ants bump against each other!

Imagine that two ants, $$A$$ and $$B$$, bump against each other at a given location of the pipe. Let us suppose that $$t_A$$ is the time it would take ant $$A$$ to leave the pipe if it were alone in the pipe, and similarly for $$t_B$$ and ant $$B$$. When the two ants $$A$$ and $$B$$ bump against each other and turn around, ant $$A$$ is now facing in the direction that $$B$$ was facing, and therefore has to walk whatever $$B$$ was going to walk, therefore the time that the ant $$A$$ is going to take is now $$t_B$$. Similarly, when the ant $$B$$ turns around and faces the same distance that $$A$$ was going to walk, the time it will take ant $$B$$ to leave the pipe becomes $$t_A$$.

When two ants bump, they swap the respective times they would take to leave the pipes if they were alone in the pipe. This shows that the ants bumping against one another does not make them spend more time in total, it just means that the ants swap the roles of the ants that were closer to the exit and farther away from the exit.