# A neat unsolved problem in number theory that kids can explore

Yesterday I saw a really neat thread on the Collatz conjecture from Alex Kontorovich

In that thread is a blog post by Alex’s friend Igor Park and Park’s blog post as a link to a neat set of lecture notes by Barry Mazur. AND, in Mazur’s notes is this “new to me” unsolved problem in number theory:

Instead of continuing on our journey through Mosteller’s 50 Challenging Problems in Probability, I decided to explore this problem with the boys today.

Here’s the introduction to the problem and a bit of playing around with a few of the small cases:

In the last video the boys thought that the squares would all have to be odd and the primes would have to be odd. Here we explored both of those conjectures. That exploration led to a discussion of why odd numbers always have squares that are congruent to 1 mod 8:

Now we continued the discussion from last video and investigated the primes that could appear in this problem. We started by showing that 2 could never appear and then eventually found that only primes of the form 4k + 1 could appear:

Next we moved to the computer to explore more cases of the conjecture. This was mainly an exercise into writing a simple program in Mathematica, but it led to an interesting discussion as well as an idea for further exploration:

Finally, we modified our program to explore the number of different solutions to the problem for each number. The modification to the program was actually really easy and the histogram was fascinating to see:

It is really fun to be able to explore an unsolved problem with kids. I especially love unsolved problems that allow kids to get in some secret arithmetic practice will getting a bit of exposure to some advanced ideas in math. Seeing this problem yesterday and getting to explore it today with the boys was a real treat!

# Having the boys talk through Problem #2 from Mosteller’s 50 Challenging Problems in Probability

This fall we are going through Frederick Mosteller’s 50 Challenging Problems in Probability. Our first project was last week, today we tackled problem #2. The problem goes like this:

You are challenged to win two games in a row in a series of three games against opponent A and B. B is a better player than A. You can choose to play the the three games in order ABA or BAB – which order gives you a better chance of winning two games in a row?

Here’s the problem and some initial thoughts from the boys – they found the problem to be pretty challenging:

Next we chose some specific probabilities of winning the games against player A and against player B and calculated the exact probability of winning two games in a row in each case:

Having worked through a specific calculation, next we solved the problem in general:

Finally, we went to Mathematica to run some simulations and see if our results matched with theory (and also to introduce the boys to some basic logical operations in programming):

# Follow up #2 to John Shonder’s US weather data visulaization

Two weeks ago I saw an amazing piece of work by John Shonder shared on Twitter:

I’ve already done two projects with the boys using Shonder’s ideas. The first was just walking through his code and showing him that the underlying ideas weren’t that complicated:

Using John Shonder’s Amazing US Temperature visualization wtih kids

At the end of that project I asked the boys for follow up ideas. My younger son (in 7th grade) thought it would be interesting to look at percent change rather than raw temperature change. We did that follow up yesterday:

Follow up #1 to John Shonder’s US temperature change visualizaiton

My older son (in 9th grade) thought it would be interesting to see if we could use the data to make predictions about future temperatures. We looked at that idea today.

Since an even cursory discussion of predictions is way more complicated than I’d like a 15 min talk with a 7th grader and an 9th grader to be, I decided to focus more on best fit curves rather than on actual predictions.

A funny side note to this discussion is that when I told my older son about this change he said – “That sounds pretty hard.” I told him not to worry, that there was a Mathematica command that does the fitting. His response was “of course there is” – ha ha.

So, we started today’s project by looking at plots of some of the county average temperature data. One thing I did here was have the boys estimate what a best fit line would look like by placing a ruler on the computer screen:

Next we used Mathematica to find the best fit line to the data and used Shonder’s code to do a county by county visualization of the slope of that best fit line.

Not too surprisingly, this visualization looked a lot like Shonder’s original one and the percent change one we looked at yesterday. The fact that all three of these visualization looked pretty similar led to a nice discussion about why that wasn’t so surprising:

Next we fit with a quadratic function rather than a line. As with the fit to the line, we looked a several counties first to get a feel for what was going on:

Finally, we did a county by county visualization of the $x^2$ coefficient of the quadratic polynomial. Here we got a visual that looked very different from the ones we’d seen before:

I’ve really enjoyed the discussions that we’ve had using Shonder’s project. It is amazing to me how Mathematica (and Shonder’s terrific code!) makes a pretty difficult data analysis project accessible to kids.

# Follow up #1 to John Shonder’s US temperature change visualization

Last weekend we did a project inspired by this incredible data visualization project from John Shonder:

That project is here:

https://mikesmathpage.wordpress.com/2019/06/16/using-john-shonders-amazing-us-temperature-visualization-with-kids/

At the end of last week’s project I asked the boys to think of some follow up projects. My younger son thought it would be interesting to see the percent change in temperature rather than the absolute difference. We did that project today.

The boys have been hiking in the White Mountains for about a week and just got home last night. So, to start today’s project we took a quick look at last week’s project and talked about what changes we’d need to make to implement my younger son’s idea:

Off camera the boys looked up how to convert Fahrenheit to Kelvin so that we could talk about percent change. We started the second part of today’s project by looking at the code where Shonder takes the difference between 10 year averages and changing that code to compute the percent increase.

It is great that Shonder’s code is so accessible that we can make this simple change and spend time talking about math that is easily accessible to a 7th grader.

To finish, we took a careful look at the new visualization. For clarity, below the video are the pictures from last week and this week. I should have prepared both of these for the boys to see in the video, but even though I didn’t, their thoughts on the change are really interesting:

Here’s last week’s visual:

And here’s this week’s – you have to look pretty carefully to see the differences, but I still think today’s project was worthwhile:

# Using John Shonder’s amazing US temperature visualization with kids

The videos in this project are a bit longer than what we normally do. Also the 2nd one is badly out of focus even though I didn’t do anything that I know of (!!) with the camera between any of the videos. Oh well, don’t let the length or the focus issues distract from Shonder’s amazing piece of work.

So, last week I saw a really neat tweet about a blog post on Wolfram’s site:

I started the project by showing the boys Shonder’s visual and asking them what they thought about it and what they noticed. At the end I showed them the raw data and we talked about some of the difficulties that come when you are dealing with a big data set:

Next we walked through Shonder’s blog post. I wanted to show the boys that although some of the code looks a little complicated, for the most part Shonder was dealing with ideas that were reasonably easy to understand. So, almost all of the steps and ideas in this presentation were things that were accessible to kids.

Next we stepped through the individual lines of code using our home version of Mathematica. Here we go pretty slowly and carefully through most of the code and discuss (and show) what each command does to the data. I hoped that this slow walk would help the kids see that although the pieces of the code might have looked a little intimidating, it was mostly pretty simple stuff. Happily, the boys seemed to understand almost all of the steps, which was really fun!

Finally, I asked each of the boys to think (off camera) of a follow up project that they thought we could do.

My younger son thought about making a graph showing the percent change in the average temperature. That led to a short discussion of how we’d measure that percent change, which was nice. This idea seems like one that we can implement pretty easily and should be accessible for a 7th grader.

My older son wondered if we could make a prediction about future temperatures. This idea is obviously quite a bit more difficult, but hopefully we can find a way to explore it. One thing that might be fun would be to take the first 50 years of data, use that for a prediction of the next 50 years, and then compare that prediction to what actually happened.

Anyway, we’ll think about how to explore both of the ideas in the next week:

I really had a lot of fun prepping for this project and talking about the ideas (and the implementation in Mathematica) with the boys today. It is really amazing to me that data analysis ideas like the one Shonder is sharing here can be made accessible to kids.

# Playing with an amazing program on “Scissors Congruence” shared by Francis Su

I saw an incredible tweet from Francis Su yesterday:

After exploring the program a little bit last night I thought it would be really fun for the boys to play with it this morning. So, I showed them the basics of how the program works and had them each play around for 10 min. Here are their thoughts:

Older son next (in 9th grade):

I am really happy that this program won an NSF award – what an incredibly fun way to share an advanced math topic with everyone!

# Talking primes using Dirk Brockmann’s “Prime Time” explorable

I’ve been a huge fan of Dirk Brockmann’s explorable math activities since I first learned about them. The full list is here:

Dirk Brockmann’s Explorables

Today’s project was inspired by the “Prime Time” program – direct link here:

Dirk Brockmann’s Prime Time Explorable

I started the project today by asking my son to tell me some things he knew about primes. He gave the definition of a prime numbers, explained how we know that there are infinitely many primes, and talked about twin primes, though he apologized for not knowing how to prove that there were infinitely many twin primes:

Next I showed him the polynomial $n^2 + n + 41$ and we talked about this equation producing a lot of primes.

Now we went to the “prime time” explorable and my son talked about what he saw in the first two examples -> the Ulam spiral and the Sack spiral.

Finally we looked at the last two patterns -> the Klauber triangle and the Witch’s spiral.