I was looking for a relatively stress-free project to do with the boys this morning and thought it would be a good day to review one of my all time favorite math projects for kids -> Larry Guth’s “no rectangles” problem. One of my all time favorite moments doing math with kids came when I used this problem for a 2nd and 3rd grade Family Math night at my younger son’s elementary school a few years ago. The problem is accessible to kids of all ages and also of interest to research mathematicians.

We started with a quick review (and lucky clarification!) of the problem and then the boys tackled the 3×3 case:

Next we moved on to the 4×4 case. The thought process the boys went through here I think shows why this is such a great math problem for kids to talk through:

Next I had a film goof up – luckly it was just 30 seconds of introducing the 5×5 case and telling them that answer to the 5×5 case was 12 squares. Don’t know what happened to this piece of the film, but since the plan was for the boys to play with this part off camera anyway I didn’t bother trying to fix it.

In any case, here’s the 12 square covering for the 5×5 they found and then a brief discussion about the surprise that comes when you move to the 6×6 square:

Again, this is one of my favorite math projects for kids – and kids of all ages. It is a really fun problem to play around with.

3 thoughts on “Revisiting Larry Guth’s “no rectangles” problem – one of the best math activities for kids I’ve seen”

Do you have a link to somewhere I could read about the 6×6 and NxN cases? I found Larry Guth’s lecture announcement, but I was unable to find an examination of the NxN problem.
Thanks for the interesting problem to think about!

Do you have a link to somewhere I could read about the 6×6 and NxN cases? I found Larry Guth’s lecture announcement, but I was unable to find an examination of the NxN problem.

Thanks for the interesting problem to think about!

The only reference I have is from the integer sequence database:

https://oeis.org/A072567

I learned about the problem from attending Larry Guth’s lecture

Thank you!