How to solve it? Let S be the number of squares the box has to travel before it hits the corner.

The box hits the top after 19 squares, and every 38 squares thereafter. So S = 19 + 38a = 19(1 + 2a).

It hits the side after 31 squares, and every 62 squares thereafter. So S = 31 + 62b = 31(1 + 2b).

We need whole-number solutions to 19(1 + 2a) = 31(1 + 2b). Both are prime, so 1 + 2a = 31 and 1 + 2b = 19. Then a = 15, b = 9, and S = 19 x 31 = 589.

Answer: the box hits the corner after travelling 589 squares, or 5890 pixels. On the way it bounces off the horizontal edges 31 times, and the vertical edges 19 times.

If the box is travelling at 57 squares every 10 seconds (by inspection), then it will hit the corner after 1:43.33333…. seconds.

I’m preparing to teach math and I don’t mind checking my understanding (or naïveté) publicly, in front of this legion of esteemed commenters; I want to know that I’m seeing the physics correctly. The logo will never hit the corner for the same reason that it never has, right? Once hitting the corner means forever hitting the corner. Is getting to that aha-point one of the goals of the lesson being considered here?

Yes, of course. Only if the rectangle is a square would a logo that hits the diagonally opposite corner on its maiden voyage be eternally committed to that one reflective path. I should check the shape of the box that I’m trying to think outside of!

Update I made the python program and I feel glorious. The parameters are simple provide a width and height and starting width and height and the program runs until it hits a corner. Then a grid pops up of your specified width and height and shows all the squares that were covered with a color coding (green is the initial square, orange are passing squares and red is the final square).

You can download it here.

It’s written on python 2.5.x and might not work on python 3.x.
The four variables to set are at the top of the file (lines 6 – 11)

Also, I should mention a more esoteric and colorful modification of the above musing.

You are standing in a rectangular room whose walls are covered in mirrors. There is a guard with a laser gun standing in this room with you and they seem rather mad at you. They shoot you and you die: the end. Suppose you could have built some number of infinitely thin columns in this room (and suppose that you and the guard are essentially points). How many of these infinitely thin columns do you need to erect to keep the guard from being able to shoot you? (note that if the walls were not mirrors, one column, directly between you and the guard would suffice to block all possible shots)

thanks for doing a lot of the leg work. i don’t have all the technical skill to set up the short clips. how did you learn how to do this? I’d like to be able to do this on my own.

Follow up to Sue’s post:
Great that you mentioned Harold Jacobs and his book: Math: A Human Endeavor. I was Dan Meyer’s age when I discovered it back in 1972. It changed my teaching from deliverer of content to explorer of what’s possible. In fact Harold was one of the pioneers of this WCYDWT theme. Instead of asking what could be done with a potentially powerful idea, he thought about the best to do it so that it would engage his high school students in the hum-drum topics of math by using engaging scenarios and stories to challenge them. The best part of the book was not the book itself but his teacher guide – at least for his first edition – which is what I used. It took me on an exciting journey through math and sharing it with my students was absolutely wonderful. (Though intended for HS, I used it with 7th/8th graders in a private school in NYC.)

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

Alex has described the general solution.

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.

Wow – great lesson and approach. Any way you could make the after effects file available?