After seeing Phelps’s tweet I thought it would be fun to see if the boys remembered how to find the volume of the shape. So, I built one out of our Zometool set and asked them what they knew about the shape.

Here’s what my older son had to say:

Here’s what my younger son had to say:

I’m glad I saw Phelps’s tweet – it was fun to revisit some of these old projects occasionally. Also, it was a nice reminder of how easy it is to share 3d shapes with kids using a Zometool set.

Following those projects I thought it would be neat to let the computer run and find the numbers that could be written as a sum of two primes in many different ways (specifically, in more ways than any number less than it). Looking at those results produced a nice surprise:

A double surprise was that Numberphile had just (the day before) published a follow up Goldbach Conjecture video that talks a little bit about the idea that explains the pattern I was seeing:

Last night I walked the kids through some of the ideas. We first watched the end of the Numberphile video and then talked about it briefly.

Also, I was pretty under the weather yesterday, so sorry for the low energy from me in this project:

Next we moved on to looking at the Goldbach Comet and told them about the project I was looking at while they were up in New Hampshire hiking.

They noticed the same pattern that I saw and I showed them the prime factorizations of a few of the numbers on my list.

After we talked about the factoring, I wanted to show them another surprise – the Goldbach Comet looks surprisingly symmetric around the numbers that can be written as the sum of two primes in lots of ways.

Fianlly, we wrapped up the project by looking at the symmetry I mentioned above a bit more carefully. I’d like to explore this symmetry a bit more myself!

We’ve really had a fun set of projects on the Goldbach Conjecture. It is definitely accessible to kids and a great way to show them an unsolved problem in math!

My wife and kids were hiking up in New Hampshire for the weekend and I just let a simple Goldbach Comet program run in Mathematica while they were gone. Here’s a version of that program. The graph shows the number of ways to write the even integer 2N (up to 100,000) as a sum of two primes. I forgot to label the x-axis correctly, which is why the last label is 50,000 rather than 100,000.

One thing I thought would be fun to do was to look at the numbers that can be written in many different ways as the sum of two primes – so the very top of this graph. I got an bit of a surprise:

So, what this picture is showing is the number with the highest number of partitions. So 330,330 sets the record with 6,181 partitions into two primes. this record isn’t broken until 351,120 which has 6,363 different partitions.

The surprise is that there are so many numbers that are multiples of 1,001. What is it about these numbers that leads to so many different ways to write them as a sum of two primes?

A look at the factorization of the last 5 numbers suggests that these numbers have really simple factorizations (to help read this chart, the first number is factored as follows:

It wouldn’t surprise me at all if there’s a relatively easy explanation for what’s going on here – but I don’t see it! Why would numbers that factor nicely have lots and lots of ways to be written as sums of two primes?

My wife and kids are going hiking today and I was looking for any fairly light project to do with the boys before they left. This morning I thought playing around with the Goldbach Comet would be a fun idea. We learned about it last week in Numberphile’s Goldbach Conjecture video:

Today’s project needs a little disclaimer . . . . Sometimes when I decide to try something at the last minute things actually work out ok. Today was much more stumbling around than usual, unfortunately. But we had fun exploring anyway.

So, we started with some simple Mathematica code to explore the number of ways to write an even integer as the sum of two primes:

I gave the boys a challenge of finding the largest even number that can be written as the sum of two primes in 6 different ways. Then we played around a bit more – stumbling around aimlessly . . . .

Finally we used a program from the Wolfram Demonstrations Project to play around with the Goldbach Comet. That project we used is here:

I mainly used the code here to ask the kids what they thought they were seeing.

So, a fun project despite the numerous stumbles. I’d actually never heard of the Goldbach Comet prior to the Numberphile video. It was neat to play with.

Tonight my older son is at a school event. That gave me time to do a fun little extension of these two projects with my younger son.

First I reviewed the original problems:

My son solved the 2nd problem above by making triangles with sides 1, and . For this part of the project I wanted to show him a different triangle that has a side length of – a 30-60-90 triangle:

Now – for a little extra fun – we made a Zometool cube. That cube shows that the face diagonal (of a 1x1x1) cube has length . It also shows that the internal diagonal has length

Here’s the surprise – if we extend basically the same geometry to 4 dimensions, we find that the “long” internal diagonal of a 1x1x1x1 cube has length 2, and that there’s a secret little 30-60-90 triangle hiding in the cube!

Numberphile released a really nice video about the Goldbach Conjecture today:

I thought it would make an excellent project with the boys even though some of the ideas involving logarithms might be over their head. So, we watched the movie and then talked about some of the ideas that caught their eye.

Next we moved on to the individual ideas. The first one was the chart that David Eisenbud made at the beginning of the video. Drawing and then filling in this chart is a nice little arithmetic activity for a kid in elementary school.

Next we talked about logarithms. I started with an idea I learned from Jordan Ellenberg’s book “How Not to be Wrong” – the “flogarithm”. That idea is to oversimplify the logarithm by defining it to be the number of digits in the number. That simple (and genius) idea really opens the door to kids thinking about logarithms.

With that short introduction I explained what the natural logarithm was and moved on to some of the properties of primes that Eisenbud mentioned in the video (after fumbling with the calculator on my phone for a minute . . . .).

(Also, I noticed watching the video just now that I forgot to divide by 2 at one point – sorry about that.)

Finally, we checked a specific example – how many ways were there to take two primes and add up to 50? This part is about as far away from the complexity of logarithms as you can get – just some nice arithmetic practice for kids.

To warp up I asked them if they knew any other unsolved problems about primes. My older son mentioned something about twin primes. I showed the boys a simple argument (fortunately quite similar to the one Eisenbud gave in the movie for why there are lots of ways two primes can add to be a given even number) for why there ought to be infinitely many twin primes.

I think that kids are going to be naturally curious about primes. The Goldbach conjecture is one of the few unsolved problems that kids can understand. It was fun to share this video with the boys tonight.

I decided to revisit a piece of that project today to show them that both of their solutions to part 2 were essentially proofs of the Pythagorean theorem.

We started by reviewing yesterday’s project:

Next we talked about how my younger son’s way of constructing the square with area three can be used to prove the Pythagorean theorem:

Finally, we looked at the slightly different way that my older son constructed the square with are 3. This approach proves the Pythagorean theorem in a different way:

This was a fun couple of projects that came from a really innocuous sounding question.

Heard a neat problem on a math podcast today which basically boils down to this question:

If I give you a square (or a bunch of squares) of side length 1, how can you make a square with area 2?

I thought trying out this question with both of the boys would be pretty fun. Here’s how it went:

(1) My younger son went first

(2) My older son went second

Next I thought it would be interesting to extend the problem a little bit and ask them to try to create a square with area 3. To my surprise this problem was significantly more difficult – the two video below are roughly 9 min each.

(3) My younger son went first:

(4) My older son went second

I was surprised at how much more difficult the 2nd problem was for both kids. I was also surprised that they approached it the same way (my older son wasn’t home when I did the project with my younger son so it really was a coincidence).

Would be fun to find some more problems like this one.

Grant Sanderson’s latest video explaining a connection between pi and prime numbers is absolutely fantastic:

This video is sort of at the edge of what kids can understand, but it was fun to explore a few of the ideas with them even if understanding 100% of the video was probably not realistic. Our project on the first 10 min of the video is here:

I intended to divide our study of Sanderson’s video into three 10 minute sections, but the second 20 minutes was so compelling that we just watched it all the way through. After watching the last 20 min a 2nd time this morning I asked the kids what they found interesting. The three topics that they brought up were:

(i) The function,

(ii) The formula for , and

(iii) Factoring ideas in the Gaussian integers

Following the introduction, we talked about the three topics. The first was factoring in the Gaussian integers. We talked about this topic in yesterday’s project, too.

Next we talked about the function. I had no idea how the discussion here was going to go, actually, but it turned out to be fantastic. The boys thought the function looked a lot like “remainder mod 4”. Why it does look like that and why it doesn’t look like that is a really neat conversation with kids.

Finally we talked through the formula that Sanderson explained for It probably goes without saying that Sanderson’s explanation is better than what we did here, but it was nice to hear what the boys remembered from seeing Sanderson’s video twice.

I love having the opportunity to share advanced math with kids. I don’t really have any background in number theory and probably wouldn’t have tackled this project with out Sanderson’s video to show me the path forward. It really is amazing what resources are on line these days!

The old project is based on Chapter 8 from this book:

Sanderson’s new video is pretty deep and about 30 min long, so I’m going to break our project on his video into 3 pieces. Today we watched (roughly) the first 10 min of the video. Here’s what the boys took away from those 10 min:

The first topic we tackled today was how to write integers as the sum of two squares. This topic is the starting point in Sanderson’s video and the main point of the project from the Ingenuity in Mathematics project. We explored a few simple examples and, at the end, talked about why integers of the form 4n + 3 cannot be written as the sum of two squares:

Next we turned our attention to the complex numbers and how they came into play in (the first 10 min of) Sanderson’s video. My focus was on the Gaussian Integers. In this part of the project we talked about (i) why it makes sense to think of these as integers, and (ii) how we get some new prime numbers (and also lose a few) when we expand our definition of integers to include the Gaussian Integers:

To wrap up I mentioned the topic from the prior project. The question there is something like this -> since counting the exact number of ways an integer can be written as the sum of two squares is tricky, can we say anything about how to write an integer as the some of two squares?

Turns out you can, and that the average number of different ways to write a number as the sum of two squares is . Pretty incredible.

[and, of course, I confused an and in the video 😦 Looking at the prior project will hopefully give a better explanation than I did here . . . . ]

I’m always excited to go through Grant Sanderson’s video with the boys. He has an amazing ability to take advanced ideas and make them accessible to a wide audience. Sometimes making the topic accessible to kids requires a bit more work – but Sanderson’s videos are a great starting point.