# 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.

# Exploring the generalized binomial theorem with my younger son

My younger son is reading Conway and Guy’s The Book of Numbers right now and one of the early sections in the book on the binomial theorem caught his attention. We talked about the binomial theorem for a bit and then I showed him a few examples of the generalized binomial theorem and he was really interested.

Today we talked about the ideas a bit more, starting with a reminder of what the theorem says:

Now we used the ideas from the last video to take a look at an approximation to $\sqrt{2}$.

Next we looked at an approximation that turns out to be one step more complicated – $\sqrt{3}$

Finally, we went back to Mathematica to look at how good the approximation from the last video was:

It was really fun showing my son some of these advanced ideas. I’m excited to explore these ideas with a few more approximations tomorrow.

# Our first project from Conway and Guy’s The Book of Numbers

My son has started reading The Book of Numbers by Conway and Guy. Today we did are first project based on some examples early in the book:

The first example we looked at were the “hexagonal” numbers. Here my son explains what those numbers are and gives a little introduction into the surprising geometric idea that helps understand these numbers:

My son had some difficulty seeing the argument from the pictures in the book, so we tried out a few examples (a few days before this project) using snap cubes. Here’s his explanation of the surprising geometry:

The next thing we looked at in the book were the “tetrahedral” numbers. The book game an amazing proof showing a formula for these tetrahedral numbers. Here he explains this clever proof:

This was a really fun project, and I’m also really happy that this book is teaching my son a bit about reading math books – sometimes even reading and understanding just a couple of pages can take time.

# Continuing our continued fraction exploration in Mathematica

Yesterday we played around with continued fractions and showed that the square root of 2 is irrational:

https://mikesmathpage.wordpress.com/2021/02/13/my-favorite-proof-that-the-square-root-of-2-is-irrational-continued-fractions/

Today we explored continued fractions a bit more using Mathematica. I started by showing my son the relatively simple commands at taking a closer look at the continued fraction for the square root of 2:

Now we explored a few other continued fractions for other square roots and looked for a few patterns – he did notice that there always seemed to be a repeating pattern:

Next we looked at pi and found a few, fun surprises:

Finally, we looked at e. We only had about 2 min of recording time left, so this last part was, unfortunately, a little rushed:

The last few days exploring continued fractions has been really fun – hoping to do a few more projects over February break studying them.

# My favorite proof that the square root of 2 is irrational -> continued fractions

Last week my younger son and I did two fun projects studying the proof in this tweet from Lior Patcher:

This weekend I thought it would be fun to explore my favorite proof – the approach using continued fractions.

We’ve talked about continued fractions before, but probably not for a few years, so I started the project today by asking my son what he remembered about them:

Before moving on to the square root of 2, we talked about why rational numbers would always have finite continued fractions:

Now we calculated the continued fraction for the square root of 2 – it has a pretty fun surprise:

Finally, and this part was just for fun, I showed him the neat little mathematical trick for quickly calculating the convergents. We looked at the first few fractions that were good approximations to the square root of 2.

# A 2nd try at looking at Tom Apostol’s geometric proof that the square root of 2 is irrational

Yesterday we did a project inspired by this tweet from Lior Patcher:

That project is here, but also it isn’t one of our best:

https://mikesmathpage.wordpress.com/2021/02/06/sharing-tom-apostols-irrationality-of-the-square-root-of-2-with-my-younger-son/

Since I didn’t think I did a great job communicating the main ideas in Apostol’s proof yesterday, I wanted to try again today. First we started with a review of the main ideas:

Next we tried to take a look at the proof through a slightly different lens -> folding. I learned about this idea yesterday thanks to Paul Zeitz. It takes a bit of time for my son to see the idea, but I really like how this approach helped us understand Apostol’s proof a bit better:

Finally, to really drive home the idea, I asked my son to see if he could see how to extend the proof to show that the square root of 3 is irrational. We were down to about 3 min of recording time, unfortunately, so he didn’t finish the proof here, but you can see how a kid thinks about extending the ideas in a proof here:

So, as I was downloading the first three films, my son continued to think about how to use the ideas to prove that the square root of three was irrational. And he figured it out! Here he explains the idea:

I’m definitely happy that we took an extra day to review Apostol’s proof. It feels like something that is right on the edge of my son’s math ability right now, and I think really taking the time to make sure the ideas could sink in helped him understand a new, and really neat idea in math.

# Sharing Tom Apostol’s irrationality of the square root of 2 with my younger son

I saw a really neat tweet from Lior Patcher last week:

I thought it would be fun to share this proof with my younger son since the geometric ideas in it are both surprising and super interesting. Unfortunately this one didn’t go nearly as well as I’d hoped. I missed a good idea that he had and got caught up in a few details that weren’t that important. Oh well, even after 10 years of doing these projects, I don’t have a good feel ahead of time for how they’ll go.

That said, here’s what we did. I started by having him walk through what is probably the most common proof that the square root of 2 is irrational:

Next we looked at Apostol’s proof and talked about some of the geometric ideas, and I just 100% missed that he was absolutely on the right track:

Now we took a look at an algebraic approach to the problem using the Pythagorean theorem. This part also didn’t go as well as I hoped and I might revisit it tomorrow just to make sure that these algebraic ideas made sense:

Finally, we came back to the geometric ideas since I realized that he was on the right track. Unfortunately I spent way too much time at the end of this part on a minor point. But hopefully the main geometric idea that we talk through in the first half of this video came through ok.

It is always disappointing when these projects don’t go quite as planned – I definitely want to push the “try again” button on this one.

# Part 2 of talking about infinity with my younger son

Yesterday my younger son and I talked about why the set of rational numbers has the same size as the set of positive integers. That project is here:

Today we followed up on that project by taking a look at why the set of real numbers is larger. First, though, I wanted to tie up one loose end from yesterday. Unfortunately, though, I left things a little too open ended and we didn’t tie up that loose end in one video:

Now that we understood that we’d over counted (in some sense) yesterday, I wanted to show him why that over counting didn’t really change the proof. I also wanted to show him one real curiosity that comes up with infinite sets:

Now we moved on to talking about real numbers. I suspected that he’d seen Cantor’s diagonal argument before (and he had), so I asked him to sketch the proof. He got most of the way there:

Finally, we tied up the loose ends from his summary of Cantor’s diagonal argument and talked about one other surprise with infinite sets.

We had a great time talking about infinity this weekend. It amazes me that kids can get their heads around ideas in math that were absolutely cutting edge just over 100 years ago!

# Talking about infinity with my younger son

My younger son is reading Bridges to Infinity right now as part of the math / reading project that we are doing in 2021.

The chapter discussing various types of infinities caught his eye this week, so today we talked about infinity. The specific goal of the project (for me, anyway) was to help him understand why the size of the set of positive integers and the size of the set of positive rational numbers was the same infinity.

We started with a quick introduction about infinity and what it meant for two infinite sets to be the same size:

Next we moved on to talking about the positive rational numbers and how to make a 1 to 1 map between them and the positive integers. This is a longer than usual video, but it turned out my son had an idea that I wanted to pursue to the end.

In the last video we figured out that if we could get a map from the integers to the rational numbers between 0 and 1 we’d be done. Here my son shows his idea for how to create that map:

This project is going to be one I remember for a long time! It is really fun to see a kid get some ideas on a really challenging math problem, and then follow them all the way through to the end!

# Part two of talking through the Bayes’ theorem chapter of How not to be Wrong with the boys

Yesterday we did an introductory project on Bayes’ Theorem inspired by chapter 10 of Jordan Ellenberg’s How not to be Wrong:

https://mikesmathpage.wordpress.com/2021/01/23/talking-through-a-bayes-theorem-problem/

Yesterday’s discussion helped the boys understand the problem that Ellenberg is discussion in chapter 10 of his book a bit better (hopefully anyway!). Today we took a crack at replicating the calculations in the book relating to the roulette wheel example.

First we revisited the example from the book to make sure we had a good handle on the problem:

Next we talked through the details of the process that we’ll have to follow to replicate the calculations that Ellenberg does. Following the discussion here the boys did the calculations off camera:

Here we talk through the numbers that the boys found off camera – happily we agreed with the numbers in the book.

At the end of this video I introduce a slight variation on the problem – instead of getting R, R, R, R, R in a test of 5 rolls, we get an alternating sequence of R and B for 20 rolls:

Here are their answers – and a discussion of why they think the answers make sense – for the new case I introduced in part 3 of the project:

This two project combination was really fun. My younger son said that he was confused by the roulette wheel example, but I think after these two projects he understands it. I think it is a challenging example for a 9th grader to understand, but with a little discussion it is an accessible example. It certainly makes for a nice way to share some introductory ideas about Bayesian inference.