If the distances the box travels horizontally and vertically (call them “h” and “v”) are relatively prime, then the box will always hit a corner eventually.

If the distances are not relatively prime, then it will eventually hit a corner if and only if it starts at a multiple of gcd(h,v).

So I would suggest playing with different widths and heights. “Does the box ALWAYS hit a corner eventually?” (No; you can set up a square and have it bounce around in a diamond, for instance.)

Also, you can tile the plane with copies of the screen. Then the path of the bouncing box is isomorphic to a straight line along the plane. This is another way to arrive at the same formulation; you are just asking whether the line passes through a multiple of (h,v). This is arguably an easier way to formulate the solution if the box does not move at 45 degrees.

Lots to play with here, although only your most advanced students are likely to “get” the complete solution.

I participated in Sue’s effort to help homeschooling parents learn the book. She did chapter’s 1 and 2 and I did chapter 3
For those interested in more details, here was my take on Chapter 3- Functions and their Graphs
Link

http://illuminations.nctm.org/activitydetail.aspx?id=28

I loved this problem, because it connects so many things.

Best,
Sam

The chapter is titled Mathematical Ways of Thinking. I once led an online group using this book. (We fizzled out after the first chapter.) My notes on this problem are here.