# Working through a neat circle / rectangle puzzle from Catriona Agg

Earlier this week I saw a really neat puzzle from Catriona Agg:

We’ve done lots of projects with Catriona’s puzzles in the past, so just search for “Catriona” and you’ll find them.

My younger son spent some time off camera solving the puzzle and then I asked him to walk through his solution. His solution gets the main idea about tangents and circles, and then computes the radius of the semicircles using the Pythagorean theorem:

Typically when we play with one of Catriona’s puzzles I have my son look through the twitter thread afterwards and find a neat solution. I took a different approach today and showed him how to use similar triangles to get to the answer with slightly less computation:

I really like Catriona’s puzzle. I also think that my son’s explanation is a great example of what kids doing math looks like.

# Showing that 1/40 is in the Cantor set is a great arithmetic exercise for kids

Yesterday we did a fun project on the Cantor set inspired by an amazing tweet from Zachary Schutzman:

That project is here:

Today we extended some of the ideas from that project by showing that the number 1/40 is in the Cantor set. Here’s how my son approached the problem – the idea he uses builds on the idea we talked about with the number 1/10 in yesterday’s project. I was happy to see that those ideas had stuck with him!

Now that we knew 1/40 was in the Cantor set, we talked about what other numbers of the same form must be in it. Although we don’t prove it (that’s what the paper in Schutzman’s tweet does), he’s now found all of the numbers with finite decimal expansions that are in the Cantor set

Finally, I wanted to go down a path relating these base 3 expansions to infinite series, but my son’s ideas took this last part in a slightly different direction. Which was fine and also fun. It really shows that kids can have fun exploring – and also have the capacity to have some great ideas about – infinite series.

These two projects have been really fun. I think the ideas about the Cantor set are great for kids to play around with!

# A fun fact about the Cantor set and a great arithmetic exercise for kids!

Yesterday I saw an amazing tweet about the Cantor set:

The amazing paper posted by Zachary Schutzman was in response to this question posed by Jordan Ellenberg:

I thought explaining some of the ideas about the Cantor set to my younger son and then having him play around with some fractions in base 3 would make a pretty fun project. So we tried it out tonight.

First we talked a bit about the Cantor set and he shared some initial thoughts:

Next I asked him to try to compute 1/4 in base 3. I always like projects like these with kids as they sneak in a little extra practice with fractions. Here’s his work:

Finally, I asked him to compute 1/10 in base 3 using an idea I mentioned at the end of the last video. After he did that, I asked him to find a few other fractions of the form k/10 that must also be in the Cantor set.

This was definitely a fun project. The math ideas here are slightly tricky, but hopefully the work here shows that the are accessible (and interesting!) to kids.

# Talking about the Gaussian Integers with my younger son

My younger son is reading The Book of Numbers by Conway and Guy right now:

Last night he read about the Gaussian and Eisenstein Integers. Today we talked about the them (well, just Guassian integers).

Here’s the introduction to what they are in his words:

Before diving into the Gaussian integers, I asked him to give some of the ideas he thought were important in the regular integers. The talk here gave us some good things to talk about in the next video:

Now we talked about Gaussian integers – one fun thing he talked about here was that multiplication by i was the same as rotating by 90 degrees:

To wrap up today’s project we talked about why the number 5 is not a prime in the Gaussian integers. It was nice that he was able to show this!

I’m excited to introduce the Eisenstein integers tomorrow – hopefully today’s project helps prepare for the slightly more complicated math we’ll see there.

# Continuing our exploration of Collatz-like sequences

Yesterday my younger son and I did a fun project inspired by this blog post from Stephen Wolfram:

That project is here:

https://mikesmathpage.wordpress.com/2021/03/06/playing-with-the-collatz-conjecture-inspired-by-a-stephen-wolfram-blog-post/

Today we continued our exploration and started with a quick look at the 5n+1 problem we finished with yesterday:

Next we moved on to the mod 3 version that Stephen Wolfram looks at in his blog post.

Finally, we looked at what happens for the number 469 in this version of the Collatz conjecture, and talked through some ideas about how we would study if it ever repeats.

This was a fun exploration with my younger son. Wolfram’s blog post is incredible, but also accessible to kids. I love being able to explore ideas like these with kids.

# Playing with the Collatz conjecture inspired by a Stephen Wolfram blog post

I saw a really neat blog post from Stephen Wolfram last week:

The blog post has so many different ideas that you could share with kids, but I decided to spend the weekend exploring various versions of Collatz-like sequences with my son.

We started by looking at some simple code in Mathematica to generate Collatz sequences:

Next we looked at how long it took various numbers to get to 1 in the Collatz sequence and looked at a histogram of the numbers. We got a fun surprise:

We wrapped up today’s project by looking at what happens when you replace the 3x + 1 rule in the Collatz conjecture with a 5x + 1 rule. I don’t remember ever seeing this idea before and it was one that really surprised me reading through Wolfram’s blog post this morning:

Tomorrow we’ll extend today’s project by looking at the 4x + 2 and 7x + 1 version that Stephen Wolfram mentioned in his tweet. Hopefully that’ll make for a really fun project, too.