How to view two small balls bumping into each other as passing through each other(2016 Pascal Contest Q24)?

Question

Here is a question from 2016 Pascal contest(Waterloo, Canada). The question and its full solution are accessible via https://www.cemc.uwaterloo.ca/contests/past_contests.html

Please find below screenshots for quick references.

I couldn't understand the following two parts and look for some help:

1. How to view two small balls bumping into each other as passing through each other?
2. How does the configuration provided in the solution preserve the orginal motions and work out so well?

Thank you!

Answer 1

As it says in the question, when two balls collide they both swap directions but keep moving at the same speed. So in other words, a moment before the collision you have one ball going $1\ ms^{-1}$ to the right and one ball going $1\ ms^{-1}$ to the left. Then a moment after the collision you have one ball going $1\ ms^{-1}$ to the left and one ball going $1\ ms^{-1}$ to the right - so if you can't tell which ball is which then nothing has changed.