I know you weren’t asking, but I see n overlapping squares.

— Hunter Patton (@professorpatton) November 12, 2019

What about you? How do you see the pattern in the tweet above growing? Please take a look before I completely ruin it for you. (So much for my rule of “never tell an answer.”)

I loved that Hunter saw overlapping squares in pattern #307 from Caden Glover. I saw the pattern like this instead — with the constant four-circle square, and groups of three circles wrapping the upper right corner, and the number of groups is one less than the step number.

Some 30 hours later, I was at my desk creating another pattern, and I started out with a diagonal of increasing cubes (highlighted in yellow), and above and below this diagonal are more cubes (marked in purple). Therefore, for any step *n*, I see the middle as (*n* + 1) and the purple ones are two equal groups of *n*.

It was not until I finished creating step 3 that I realized I’d made the same pattern as the one from Caden. I didn’t want to scrap it because I didn’t mean to copy his pattern and really had built it from “scratch,” and now it is pattern #324 on visualpatterns.org.

It might be my favorite one thus far because I can see it in different ways. Here are the overlapping squares that Hunter saw:

What’s fun is sometimes I don’t see the pattern in its “simplified” form until I’ve simplified the equation. (I have to do this for the answer key). The number of cubes *C* is related to the step number *n*, such that *C* = 3*n* + 1.

And I wouldn’t be me if I didn’t always try to see a rectangle in any pattern. (The green rectangles have dimensions of 0 by 3, 1 by 3, and 2 by 3 for steps 1, 2, and 3, respectively.)

Many of my students will try to see if the entire step can be enclosed in one rectangle, then minus the negative space. The negative space (missing cubes) of this pattern is fun to discover too!

Do you see another way?

Six years ago when I created the site I had hoped to have 180 patterns to match the number of school days. We are now at 324! I’m so grateful for all the pattern submissions and for all the ways that the site gets shared.

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