Revisiting Larry Guth’s “No Rectangles” problem

Yesterday I saw a discussion on twitter about advanced ideas in math that are accessible to kids and that do not require arithmetic. I hope to be able to write a blog post with a collection of problems meeting that description, but for now that discussion inspired me to revisit Larry Guth’s “No Rectangles” problem with my younger son today.

Larry Guth is a math professor at MIT and I first learned about the problem (introduced in the first video) in a public lecture he gave a few years ago. I’ve used the problem with kids as young as 10. In fact, it generated so much excitement from the 10 year olds that I couldn’t end that “Family Math Night” because the kids didn’t want to leave without figuring out the 4×4 case 🙂

Today we just talked through the 3×3 case. I think the three videos below do a great job highlighting how a kid can approach the problem, and also (importantly!) how this problem is accessible to kids.

Here’s the introduction to the problem and some initial thoughts that my son has:

Next we discussed a few of the situations in which my son was able to put X’s in 6 squares without getting a rectangle and how we could determine if 6 was actually the maximum.

Finally, we wrapped up the project today by looking closely at the difference between a situation in which he could only fill in 5 squares and one in which he could fill in 6. That comparison helped him see why you could never get 7 in a 3×3 grid without generating a rectangle: