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!

One more trip through Goldbach’s comet with the kids

We’ve now down a couple of projects on the latest Numberphile video on the Goldbach Conjecture:

Sharing Numberphile’s Goldbach Conjecture video with kids

Exploring the Goldbach Comet with kids

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:

An unexpected surprise for me in the Goldbach Comet

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!

Exploring the Goldbach Comet

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:

Our first project from that video is here:

Sharing Numberphile’s Goldbach Conjecture video with kids

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:

The Goldbach Coment on the Wolfram Demonstrations Project site

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.

Sierpinski Numbers

I was trying (unsuccessfully) to track down a reference on the chaos game for Edmund Harriss and ran across an unsolved problem in math that I’d never heard of before -> the Sierpinski Numbers.

Turns out that Sierpinski proved in 1960 that there are infinitely many odd positive integers $k$ for which the number:

$k * 2^n + 1$

is not prime for any positive integer $n$.

It turns out that the smallest known Sierpinski number is 78,557, though there are 4 smaller numbers for which no primes have been found, yet. Those numbers are 21181, 22699, 24737, 55459, and 67607.

There’s lots of info on the Sierpinski numbers on Wikipedia:

Wikipedia’s page on the Sierpinski numbers

Tonight I wanted to explain a bit about the Sierpinski numbers to the boys as a way to review modular arithmetic. I also thought it would be interesting to see how they thought you could attack a problem like this one – especially in the 1960s!

So, here’s how we got started – a bit of Sierpinski review and then an introduction to the theorem mentioned above. It isn’t the easiest thing for kids to understand, so I wanted to be extra sure they understood all of the parts:

Next we talked a bit about modular arithmetic and why it wasn’t too hard to see, for example, that lots of the number we were looking at were divisible by 3. The math work here is a great introductory modular arithmetic exercise for kids.

Next we went to Mathematica to explore the modular arithmetic a bit more. Once we had the idea with 3, it was a little easier to see why there were repeating patterns with the remainders mod 5. The fun part was that the boys were able to see that one out of every 4 numbers would be divisible by 5.

Finally, we looked at the problem a slightly different way and tried to see if it was easy or hard to see if 3 (or 5 or 7 or 9) was a Sierpinski number. Would we ever see primes?

This project was really fun – it is always neat to stumble on an unsolved problem that is accessible to kids. Also, I’d really love to know how Sierpinski’s proof went – sort of amazing that it took 8 years after the proof that there were infinitely many numbers with this property to find the first one!

Sharing Numberphile’s Collatz Conjecture video with kids

Numberphile published a beautiful video on the Collatz Conjecture today. I thought it would make for a fantastic project with the kids tonight:

We have looked at the Collatz Conjecture before, so we aren’t starting from scratch here. Two of our prior projects are here:

Revisiting the Collatz Conjecture

the Collatz Conjecture and John Conway’s Amusical Variation

I started the project tonight by asking the kids what they thought was interesting about the video:

Next we tried to recreate the “tree” that was in the video. This exercise was a nice way to check that the kids understood what was going on in Numberphile’s video:

To wrap up I wanted to walk through one example of how the Collatz conjecture plays out. Somewhat unluckily, though, my son chose 31 as the starting point. 31 takes more than 100 steps to converge!

BUT, this video shows why I think the Collatz conjecture is such a fun math idea to share with kids – you can sneak in a lot of arithmetic practice ðŸ™‚

So, we gave up after maybe 30 steps in the last video and went to check how long it would take to converge using Mathematica. Someday I’ll learn that when I zoom in too far on Mathematica the video gets super fuzzy . . . but today was not that day ðŸ˜¦

I’m really grateful to Numberphile for their video – I think videos like it will really help show off the beauty of math to a large audience.

Sharing Kelsey Houston-Edwards’s philosophy of math video with kids

Kelsey Houston-Edwards is making a series of math videos and the first two are outstanding. We looked at the first one last week:

Sharing Kelsey Houston-Edward’s video with kids

This week’s video is about philosophy and math. A deep subject, for sure, but one which the kids thought was interesting. Here’s the video (and the twitter link so you know when the new videos appear!):

Here’s my older son’s reaction and a few things he thought were interesting:

and here’s what caught my younger son’s eye:

It is so great to see someone doing such an incredible math outreach program. I’m so excited about this video series!

Explaining how 1 + 2 + 3 + . . . can possibly equal -1/12 to a kid

When I did the my biographies for my kids last week my older son said that the thing in math that he’s see but that he does not believe is this equality:

1 + 2 + 3 + 4 + . . . . = -1/12

This sum was made popular by a Numberphile video a couple of years ago (which now has over 4 million views!):

Â

there have also been several good follow ups. For example this video with Ed Frenkel which was also produced by Numberphile:

Â

and this video by Mathologer which is absolutely excellent:

Â

I spent some time today trying to think about how to discuss this series with my older son. I’m glad that he is bothered by the result – it is obviously very very strange. Obviously I can’t go into the details about the Riemann Zeta function with him, but I still think there’s some what to help him make some sense of the series. So, I spent the day reviewing some ideas in G. H. Hardy’s book “Divergent Series.” Here are a few passages that caught my eye:

(a) Book Cover

I don’t remember where I heard about this book. My best guess is that it was mentioned in Jordan Ellenberg’s “How Not to be Wrong” in the section about Grandi’s series. Unfortunately I only have the audiobook version of “How not to be Wrong” and don’t know how to search it!

(b) first passage

The remark beginning at “It is plain . . . ” caught my attention.Â  This is right at the beginning of the book – section 1.3.Â Â  The statement:

“it does not occur to a modern mathematician that a collection of mathematical symbols should have a ‘meaning’ until one has been assigned to it by definition.”

also felt very powerful to me.

(c) second passage

The continuation of the previous page is also important – the point about Cauchy was definitely mentioned in “How not to be Wrong” as well.

(d) third passage

Â

For the third passage we have to go much later in the book – nearly to the end, in fact.Â  The passage here – 13.10.11, in particular – shows the strange result.Â  Not in a Numberphile video, or some other internet video, but in a math textbook by G. H. Hardy:

(e) fourth passage

Finally – and this really is just about the last page of the book – section 13.17 provides a word of caution and an example of what can go wrong playing around with these divergent infinite series.

Â

So, I’m going to spend the next few days and maybe even the next few weeks thinking about how to share some sort of idea about this strange series with my son.Â  I’ll welcome any suggestions!

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 ðŸ™‚

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.

MoMath’s “Beautiful Math” collection, and some “beautiful math” for kids

The museum of math has put together a nice (and growing!) collection of videos of mathematicians talking about beautiful math:

Watching the videos it struck me that they were aimed at adults and older kids, but I thought it would be easy to show some beautiful math that younger kids could appreciate, too. I thought of two of my favorite projects with the boys and then asked each of them to tell me what they thought was the most beautiful math that they’d seen.

My two favorites:

(1) The Chaos game:

Our project is here:

Computer Math and the Chaos Game

and my favorite part starts about 2:30 into this video and goes for about 45 seconds:

(2) John Conway’s “Amusical” version of the Collatz Conjecture

Our project is here:

The Collatz Conjecture and John Conway’s “Amusical” Variation

and my favorite part is at the end where we convert the “amusical” process to music. The music starts around 2:00 – prior to that is just explaining Conway’s Collatz process in the Mathematica code. I love it when my younger son says “I didn’t know you could hear 20”:

(3) My older son’s choice for beautiful math – the 4th dimension

We’ve done a couple of projects related to the 4th dimension – here are a few:

Carl Sagan on the 4th Dimension

Sharing 4d-shapes with kids

which had a fun connection to our Zometool / bubble project. Around 1:00 in the video is a great moment – “who knew that bubbles could find the center of a tetrahedron?”

That comment from my son led to this wonderful drawing posted on twitter:

Another really fun higher dimensional problem for kids who know the Pythagorean Theorem is a neat problem I learned from Bjorn Poonen:

Talking through Bjorn Poonen’s N-dimensional Sphere Problem with kids

(4) My younger son told me that he thought fractals were the most beautiful math that he’d seen.

We’ve done several fractal projects for kids, too. Here are a few:

A fun fractal project – exploring the Gosper curve

After this projects, were really luck to receive some laser cut Gosper curves from Dan Anderson to play with:

Dan Anderson’s Gosper Curves

Using the Koch Snowflake to introduce fractals

Using Matt Parker’s Menger Sponge video to talk fractions with kids

One of my favorite moments from these projects happens around 3:00 in the video below as the boys stretch out a 3d printed Peano curve into (nearly) a straight line:

So, I’m really happy to see the Museum of math putting together a collection of mathematicians talking about beautiful math. I can’t wait to see more videos in the collection! Hopefully some of the videos and projects above can help younger kids see some beautiful ideas in math, too.

Talking prime numbers with kids

Saw this tweet from Steven Strogatz last night:

Glancing through the book I was happy to see that it seemed to be written for a general audience. So, with nothing already played for today’s Family Math talk with the boys I thought we’d use the book for today’s project.

The first thing I did was do a quick introduction to the book. The boys have heard Barry Mazur’s name before from his “Blob Pythagorean Theorem” video:

Here’s that project:

Using Numberphile’s “Blob Pythagorean Theorem” video in a lesson

They’ve also seen Riemann’s name in a prior 3D printing project where we played with some 3d printed Riemann sums:

3d printing and calculus concepts for kids

So, to start the project I wanted to remind them that they’ve seen some of the names before and that they also already knew a little bit about prime numbers. I’d asked each of them to pick a section from the table of contents that they wanted to study. My older son picked “questions about primes” and my younger son picked “music files and prime numbers.”

First up was my older son’s choice – questions about primes. We looked at the first page of this section which mentioned several easy-to-state unsolved problems about prime numbers. We went through a few examples of each one and then I gave them a challenge problem – these examples give a great opportunity for kids to talk about math:

After that we moved to the section about “music files and prime numbers.” Unfortunately from the point of view of taking about primes, this section is setting up the next few sections on spectral theory – so no primes here. However, this section was still pretty interesting all by itself. It is a basic introduction to Fourier analysis and the kids enjoyed it. While we talked about the section on camera we even played a few notes on the piano. The video below shows how a 4th grader and a 6th grader explain Fourier analysis ðŸ™‚

This project was really fun. I’ve been spending a lot of time lately thinking about how to share math that is interesting to research mathematicians with kids. Some of those ideas are in this post from last week:

Amazing Math from Mathematicians to share with kids

Larry Guth’s “no rectangles” problem was a huge hit with the 2nd and 3rd graders at Family Math night at my younger son’s school. I’m going to try a project on the Surreal Numbers with the 4th and 5th graders next week. So, with all that as background, it was nice to see another great resource from mathematicians to use to share math with kids.