Revisiting the Collatz Conjecture

Last week we made “math biographies” for the boys:

Math Biographies for my kids

When I asked them about their favorite unsolved problem, they both mentioned the Collatz conjecture. Unfortunately they couldn’t remember the details, but that made the choice of topic for today’s Family Math project easy!

I decided to approach the problem using sound much like we did when we looked at John Conway’s version of the Collatz conjecture:

The Collatz conjecture and John Conway’s “Amusical” variation

Before diving in to the sound, though, we reviewed the details of the Collatz conjecture:

 

Next we moved to Mathematica to listen to (a version of) the sound of the Collatz Conjecture. Sadly the camera was way out of focus here. I didn’t notice until the movies were published. So, sorry about that, but at least the sound comes through ok.

 

Next I asked the boys to change the procedure a little. My older son’s suggestion was to change the procedure from “multiply by 3 and add 1” to “multiply by 3 and add 3.”

My younger son noticed from the sound that the loop didn’t start with 1 – that was really fun to hear! Maybe a good thing, too, since the video is so out of focus 🙂

 

Finally, we made one more change to the procedure – this time “divide by 2” was replaced by “divide by 2 and then add 4.” We saw some new patterns again.

 

So, I love playing around with the Collatz conjecture with kids. First, it is always really fun to be able to show kids unsolved math problems. Lior Patchter has an incredible blog post about various different unsolved problems to share with kids at each grade level if you want more than just the Collatz conjecture:

Unsolved Problems with the Common Core

One thing that is really nice about playing with the Collatz conjecture is that you get to sneak in lots of arithmetic practice.

It is also fun to turn the numbers into music just to give the kids a slightly different way of experiencing the pattern in the numbers.

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