# A terrific list of problems from Matt Enlow

Saw a great tweet from Matt Enlow today:

I decided to try out the first problem with my younger son this afternoon and we ended up having a really nice discussion. The problem is:

Is 11 the largest prime number that has all the same digits?

There’s a lot of great math ideas hiding in the problem.

Here’s how we got started – I love hearing the progression of ideas that he has all the way to the end of the movie:

By the end of the last movie he came to the conclusion that if there was a prime number larger than 11 that had all of the same digits, the repeating digit would have to be 1. So, the next thing we did was explore the first couple of numbers made with repeating 1’s:

After realizing that even 11,111 was going to be a challenge to try to factor we end upstairs to play on Mathematica. We made some pretty quick progress! Also, seeing the non-prime results on Mathematica also helped him see some patters in the non-primes that he hadn’t notices before – score one for some computer math!

Finally, I showed him the starting list on the Integer Sequence Database that shows the first few primes of the type we were looking at:

This was a really fun project – thanks to Matt Enlow for sharing this great list of problems!

# Trying to study knots with 5 crossings

We’ve been doing a little bit of work with knots lately. Today we were studying the knot with 5 crossings, and it wasn’t so easy.

I’d guess ahead of time that 5 crossing would be tricky. There’s a lot to keep track of! Even what seems like a simple task – making a knot with 5 crossings out of rope – isn’t so easy. See if you can spot the problem:

The boys didn’t notice the problem with their knot, yet, but the problem quickly became clear when they started playing with it:

So, we started over . . .

Having now made a knot with 5 crossings, we ended the project by trying to determine which of the two knots with 5 crossings that it was. We got a little bit of a surprise when it turned out that we’d actually made the mirror image of one of the knots we printed. That was an accidental good lesson, though – mathematicians consider those two knots to be the same even though they are not always the same.

# A Zometool follow up to our cuboctohedron project

Earlier in the week we studied the cuboctahedron:

That project is here:

Playing wiht the Cuboctahedron

Also earlier in the week I saw these shapes displayed in the MIT math department:

The chance encounters with these shapes this week gave me the idea to revisit them today and see if we could build them with our zometool set. The second shape, I think, is mislabled in the MIT display case – or maybe they are just using a less common name. The usual name is the icosidodecahedron, and it is also a shape we’ve seen before:

I started the project today by showing the shapes to the boys and asking what they knew about them:

Then we went to the living room to build the shapes. The only tricky part is that the cuboctahedron needs green struts. As always, the wonderful thing about the Zometool set is that you can go from seeing these shapes on a page to holding them in your hand almost immediately!

The last part of the project was building the dual shape of the cuboctahedron. I wasn’t sure if the zome set would let us do this since you can’t exactly find the center of the triangles with zome – but we did catch a lucky break! The dual is also a shape we’ve seen before 🙂

This project was really fun – exploring geometry with our Zometool set is one of my favorite activities!

# Exploring the knots with 4 and 5 crossings with kids

In our last project we explored the trefoil knot:

That project is here:

An introductory knot activity for kids

Today we moved on to the knots with 4 and 5 crossings. We started off by comparing the trefoil knot with the 4 crossing knot – what is the same? what is different?

Also – the 3 white knots (1 with 4 crossings and 2 with 5) that appear in this projects come from Mathematica’s knot data collection and the red knot (the trefoil knot) was designed by Laura Taalman.

Next I had the boys try to make a knot with 4 crossings out of a rope. It is not as simple as it seems! One nice thing about making these knots out of rope is that they also begin to discover some of the ways you can have crossings that can be undone.

Now we compared the knot we made from the rope to the 3d printed knot with 4 crossings. In the last project we had quite a lot of difficulty comparing the different versions of the trefoil knot. Here, though, comparing the knot in our rope to the 3d printed knot was not too difficult.

Finally, we wrapped up this project by inspecting and comparing the two knots with 5 crossings. It is very interesting to hear what kids see in these two knots, and also fun to hear their ideas for how you might determine that these two knots are actually different.

Definitely a fun project. I really like exploring these knots with the kids. It makes me wonder if there is a way to go to the next level and help them understand some of the ideas that help you understand when two knots are different?

# An intro knot activity for kids

I’ve been doing a lot of thinking and playing with our various 3d printed knots lately. It feels like there are lots of great projects for kids here, but I’m struggling a little to find them.

Last night I tried something pretty simple – take several different versions of the trefoil knot and have the kids try to recreate those versions with a knotted rope.

Here are the shows and some initial thoughts about them from the boys. These knots were designed by Henry Segerman and Laura Taalman:

Next we started trying to make the shapes – first the “easy” ones 🙂

Next we moved on to some more difficult shapes – in particular the 2nd one gave the boys quite a bit of difficulty. Making the connection between these two versions of the trefoil knot isn’t completely straightforward:

I let the boys try to finish making the new knot shape with the camera off. It took a few more minutes. In this video they show how to go back and forth between the two versions:

So, definitely a fun project, but a little more difficult than I expected. We’ll see how difficult playing with the knot with 4 crossings is tomorrow.

# Extending our coordinate change project with Desmos

Last weekend we did a fun 3d printing project involving changing coordinates:

Here’s the link for that project:

Exploring some fun 3d transformations

A youtube commentator – mxlexrd – made the following Desmos version of one part of the activity after seeing the project:

mxlexrd’s Desmos version of our coordinate change project

Last night I had the boys play with it since I’d be traveling for work today. My younger son went first:

Here’s my older son:

The Desmos version of our activity is really fantastic. Even if the concepts are a little bit too complicated for my kids to understand in detail, I love how easy the program made it to explore the mathematical ideas.

# Playing with the Cubeoctahedron

Last night I was flipping through the book I bought to understand a bit more about folding – Geometric Folding Algorithms by Erik Demaine and Joseph O’Rourke:

and I ran across a short note on the cuboctahedron. The boys were taking a short trip today (school vacation week!) and I was looking for a short project to do before they left – folding up the cuboctahedron seemed perfect.

Making my life much easier was a template on Wolfram’s website:

Wolfram’s folding template for a cuboctahedron

Here’s what the boys had to say after creating the shape:

After the short discussion about the shape we went upstairs to look at the shape using the F3 program. My idea for the ~10 min discussion here was inspired by a talk by Keith Devlin I saw over the weekend:

I thought that an approach similar to a game with our F3 program would help the boys create the shape.

Here’s how we got started. The F3 program allows us to create a cube and an octahedron. It also allows you to add and subtract shapes. How can we use these 4 ideas to create the cuboctahderon?

I think the video here really shows what Devlin calls “mathematical thinking.” The conversation here was really fun (for me at least!) since trying to discuss the ideas through equations would be impossible. However, the geometric ideas are accessible to the boys via the F3 program, just as the number theory ideas are accessible to kids through Devlin’s “Wuzzit Trouble” program.

I broke the discussion into two pieces – at the start of the 2nd half of the discussion we are trying to figure out how to – essentially – flip the shape inside out. My son comes up with an idea that was very different than what I was expecting, and it worked 🙂

# Playing with 3d printed knots from Mathematica

Yesterday I learned that Mathematica has a wide variety of knots that you can 3d print. We’ve done a few knot projects in the past. Here are 3 of them:

Playing with some 3d printed knots

Dave Richeson’s knotted bubbles project

I thought that actually being able to hold the printed versions of so many different knots in your hand was going to be a game changer for knot projects, though. So, I printed a few as test cases and had the boys look at them.

My older son went first:

My younger son went next – he had a couple of things to say, but wanted to point out some of the knots in Colin Adams’s book, so we cut this video a little short so that we could go find the book:

After finding the book we were trying to match one of the printed knots with the knot in the book that he had wanted to print. The knot he wanted to print had 8 crossings and the one that we thought matched it turned out to have 7. Whoops – we had the wrong knot 🙂 A good accidental lesson that comparing two knots isn’t super easy!

I’m really looking forward to trying more projects with these prints. There are a little over 30 different knots with 8 or fewer crossings. It’ll probably take a week to print them all, but that’ll be a fun collection to have for future knot projects!

# Sharing Grant Sanderson’s Fractal Dimension video with kids

A few weeks ago Grant Sanderson published this amazing video about fractal dimension:

I’ve had it in my mind to share this video with the boys, but the discussion of logarithms sort of scared me off. Last week, though, at the 4th and 5th grade Family Math night the Gosper curve fractals were super popular. That made me think that kids would find the idea of fractal dimension to be pretty interesting.

Here are the Gosper curves that Dan Anderson made for us:

We’ve actually studied the Gosper curve several times before, so instead of just linking one project, here are all of them 🙂

A collection of our projects on the Gosper curve

So, today we started by watching Sanderson’s video. Here’s what the boys had to say about it:

At one point in the video Sanderson makes a comment that fractals have non-integer dimensions. I may have misunderstood his point, but I didn’t want to leave the boys with the idea that this statement was always true. So, we looked at a fractal with dimension exactly equal to 2:

Next we looked at the boundary of the Gosper island. I wanted to show that this boundary had a property that was a little bit strange. I introduced the idea with a square and a triangle to set the stage, them we moved to the fractals:

Finally – to clear up one possible bit of confusion, I looked at a non-fractal. For this shape we can see that the perimeter scaled by 3 and the area scaled by 7. Why is this situation different that what we saw with the Gosper Island?

Definitely a fun topic and I think Sanderson’s video makes the topic accessible to kids even if they don’t understand logarithms. I’m excited to find other fractal shapes to talk about now, too!

# Exploring some fun 3d transformations

Today’s project with the boys was exploring some simple (to code!) transformations. The question was how would the shapes change under those transformations.

I started with introducing the idea in 2d. It isn’t necessarily the simplest idea, and I had no intention to go into any details. The basic question I wanted them to think about was this – would a straight line stay straight under this transformation?

Next we looked at a tetrahedron (actually two tetrahedrons) under some similar 3d transformations:

Now for the punch line – what do the same transformations do to an octahedron?

Finally, I wasn’t planning on doing this part, but to clarify some of the ideas from the first part of the project we went up to the computer to show them what the transformations did to a line in 2 dimensions:

So, I think this is a fun way for kids to explore some 3d shapes and also begin to understand a little bit about how algebra and geometry are related