# Playing with Annie Perkins’s counting problems

I saw a neat tweet from Annie Perkins last week:

Today I thought it would be fun to play around with this idea with my younger son. First I introduced the 4-person problem and let him think through it. His thought process is a great example of what a kid learning math can look like:

At the end of the last video he’d determined that there were 3 different arrangements of the 4 people sitting around the table. In this video I asked him to find those arrangements:

Next we moved to the 5 person problem:

Finally, having decided that there were 12 different arrangements with the 5 person problem, I asked him to try to write down all 12. This is a good exercise in using counting techniques to make an organized list:

Definitely a fun problem for students, and also a really nice introduction to counting and symmetry. Thanks to Annie for sharing!

# Working with the PCMI books part 2: coloring an octahedron

Last week we got the PCMI books:

Our first project involved a neat problem about understanding the number 0.002002… in different bases:

Playing around with the PCMI books

Today I was looking for another fun problem and found another problem that I thought would make a fun project:

Barbara has an octahedron, and she wants to color its vertices with two different colors. How many different colorings are possible? By “different” we mean that you can’t make one look like the other throu a re-orientation.

I started by introducing the problem and asking the kids what their initial ideas were:

They had a couple of pretty good ideas including some basic ideas about symmetry. Using those ideas we began counting the different colorings:

We counted the cases in which 3 vertices were black and 3 vertices were red. This case proved to be tricky, but going through it slowly got us to the correct answer.

Finally, as a fun little extension, I asked them to find the number of ways to color the faces of a cube with two colors. Having solved the octahedron problem already, this one went pretty quickly, and they even noticed the connection between the two problems ðŸ™‚

I like this problem. I’m glad that the boys were able to see some of the basic ideas. When you add more colors the counting gets much more difficult and some pretty advanced math comes into play. The number of colorings with “n” colors is:

$(n^6 + 3n^4 + 12n^3 + 8n^2) / 24$

The different terms correspond to different symmetries of the cube / octahedron. We’ll have to wait a few more years to cover the complete details ðŸ™‚

# Taking about Kate Nowak’s shape

Saw this neat drawing from Kate Nowak the other night:

I was interested to see if we could make the shape from our Zometool set, and . . . .

The boys really enjoyed making the shape last night and both also made several comments about how interesting it was. This morning we talked about it a bit. Both kids focused on symmetry. I spent a bit more time with my older son exploring the different kinds of symmetry, but it was great to hear what both kids had to say. It really is an amazing shape!

Younger son first:

Older son next:

This was a really fun project. The shape didn’t take that long to build, which was lucky. It is always fun to be able to pull out the Zome set to explore something that we saw on Twitter ðŸ™‚

# Using 3d printing to talk symmetry with kids

We’ve done a lot of projects relating to platonic solids and dodecahedrons in particular. A really neat fact about dodecahedrons is that you can use the verticies to put 5 cubes inside!

It isn’t just a mathematical “fun fact” either – the symmetry groups involved play roles in important mathematical theorems.

For today’s project I wanted to explore one cube in a dodecahedron and look at the relationship between the rotations of the cube and the rotations of the dodecahedron.

We started by looking at the dodecahedron by itself:

Next we moved to looking at the cube in the dodecahedron and studied what rotating the dodecahedron did to the cube:

Finally we looked at some 3d printed models that we made to see if these models helped us explore the rotations a bit more:

I was a little disappointed that I made the 3d printed models a bit too small, but I still like how this project went. I’m going to try again with some slightly larger models with my older son.

# Revisiting our Zometool Snowman

When we first moved into our house we did a couple of fun and large Zometool projects because we didn’t have any furniture ðŸ™‚

This week I saw a fun tweet from Eli Lubroff that reminded me of one of those projects:

Here’s a part of that old project ðŸ™‚

Today we revisited that old snowman and had the boys talk about each of the Archimedean solids in the shape. This is a fun project – not just because the shapes themselves are cool – but you get a nice opportunity to talk about counting and symmetry. You’ll see in the videos that my older son is a bit more comfortable with the idea, but my younger son seems to catch on by the 3rd video.

Here’s a link to all of the Archimedean solids on Wikipedia:

The Archimedean Solid page on Wikipedia

And here’s our project:

First the bottom of the snowman – the Truncated Icosidodecahedron

Next was the Rhombicosidodecahedron

Next was the Icosidodecahedron

Finally the Archimedean Solid Snowman ðŸ™‚ Two years later and he still fits!

Definitely one of my all time favorites and a really fun way to discuss counting and symmetry!

# Taking kids through John Baez’s post about the Gyroid

[sorry for no editing on this one – had some computer problems that ate up way too much time. I finished typing with 2 minutes to spare before rushing out the door.]

I saw this neat tweet from John Baez earlier in the week:

You should be able to click through to Baez’s blog post from the tweet, but just in case that isn’t working, here’s the link:

The Butterfly, the Gyroid and the Neutrino by John Baez

I spent the rest of the week sort of day dreaming about how to share some of the ideas in the post with kids. Last night the day dreaming ended and I printed a gyroid that I found on Thingiverse:

The specific gyroid that I printed is here:

Alan Schoen’s Gyroid on Thingiverse by jamesosaurus

This project connects with several of our prior projects on 3d printing (particularly the recent ones inspired by Henry Segerman’s new book) as well as projects on minimal surfaces. Though the list below is hardly complete, here are a few of those projects:

Zometool and Minimal Surfaces

Trying out 4 dimensional bubbles

More Zome Bubbles

Playing with shadows Inspired by Henry Segerman

Playing with more of Henry Segerman’s 3d Prints

Henry Segerman’s flat torus

Using Hypernom to get kids talking about math

So, with that introduction – here’s what we did today.

First we revisited the zome bubbles to remind the kids about minimial surfaces – it is always fun to hear kids describe these complicated shapes:

Next we looked at the Gyroid that I printed last night. This shape is much more complicated than the zome bubbles and the kids sort of had a hard time finding the words to describe it – but we had a similar shape (and I don’t remember why or where it came from) that helped the kids get their bearings:

So, after playing with the blue shape for a bit and seeing some of the symmetry that this shape had (yay!) we returned to the Gyroid. The boys still struggled to see the symmetry in the gyroid (which is really hard to see!) but we made some progress in seeing that not all of the holes were the same:

Finally, we turned to Baez’s article to see the incredibly surprising connection with butterflies and physics. There’s also a fun connection with some of the work we’ve done with Bathsheba Grossman’s work and Henry Segerman’s 3D printing book:

So, a fun project. I love how 3d printing helps open up advanced ideas in math to kids. After we finished the boys kept reading Baez’s article to find the connection with neutrinos – it is really gratifying to see how engaged they were by today’s project!

# Talking through some examples from Richard Stanley’s tiling presentation

I stumbled on this incredible presentation about tilings from Richard Stanley yesterday:

Richard Stanley’s presentation on Tilings

We’ve actually done many projects on tilings in the past if you search the blog – some of the fairly recent ones are:

Counting 2xN domino tilings

Learning about tiling pentagons from Laura Taalman and Evelyn Lamb

Zome Tilings

Today we looked at two fun examples from Stanley’s paper -> tiling a chess board with dominoes and tiling a hexagon grid formed by the triangular numbers with “tribones.”

First up with the chess board. The problem here is pretty famous and a really fun one to try out with kids. Just in case you’ve not seen it before and want to try it out yourself, the problem is: If you remove two opposite corners of a chess board, can you tile the remaining shape with 2×1 dominoes:

Next we discussed the problem my younger son asked about – what happens if you removed two random squares of opposite color?

Now we moved on to the tribones and the hexagon grid. Here’s a quick discussion / introduction to the problem from Stanley’s paper:

Next I intended to have them try to build the T(9) shape from the tribones, but we took a little detour first to try to figure out why building T(6) from tribones was impossible. It probably took 10 minutes for the kids to find the argument, but it is was fun work. I wish I had left the camera running for it, but I didn’t. Here’s a short summary of the argument:

Finally, we wrapped up the project by trying to construct T(9) from the tribones:

So, a really fun weekend of tiling. I’m really happy that I stumbled on Stanley’s presentation yesterday!

# Extending our project with Ann-Marie Ison’s art

Yesterday I saw a series of tweets from Ann-Marie Ison that just blew me away. Here’s one:

I used them as a starting point for an incredibly fun project with the the boys last night:

Using Ann-Marie Ison’s increddible math art with kids

This morning the kids wanted to talk more about the circles, and – happy accident – I got this note from Ison overnight:

This program was exactly what I was looking for. I had each of the boys play with it. My older son went first:

My younger son went next. I’m sorry this video runs a little long – I just couldn’t stop. He was so engaged by the program and kept finding new interesting things to talk about:

These multiplication graphs make for one of the most interesting ways for kids to talk about math – from number theory to geometry – that I’ve ever seen.

Also, I’d point out that the math involved here goes up pretty high. Here’s a famous paper by Bjorn Poonen and Michael Rubenstein from 1997 which solved the problem of how to count the number of times the diagonals of a regular polygon intersect.

The number of intersection points made by the diagonals of a regular polygon

I’m guessing that counting the intersection points inside of the shapes in this project (as a function of N and M) is probably an extremely difficult problem.

# A second example from tiling the Aztec diamond

Yesterday I learned about the Arctic Circle theorem and used it for a fun talk with the kids:

The Arctic Circle Theorem

This morning we had a fun little coincidence as one of the problems that my son was working on was the proof that $1 + 2 + 3 + \ldots + n = (n)(n+1) / 2$.Â  The coincidence is that the number of different tilings of the nth Aztec Diamond is $2^{(n)(n+1)/2}$, so for a quick project this morning we looked at the sum and then tried to find the 8 different tilings of the level 2 Aztec Diamond:

Part 1 is a short discussion of the sum:

Part 2 is looking at the tilings of the Aztec Diamond – counting the number of tilings of the level 2 diamond is a pretty good challenge for kids.

So, a lucky second project with the Aztec Diamond. I definitely want to think more about how to share the ideas in the Arctic circle theorem with kids. I think the ideas here are something that kids will really love.

# The Arctic Circle Theorem

Today I went into to MIT and totally by accident learned about a really cool idea in math that I’d never heard of before – the “Arctic Circle Theorem.” Even more fun, it turns out that one of the mathematicians who proved the theorem is Jim Propp whose amazing blog Mathenchant has already inspired a couple of our projects.

Way, way, waaaaaaay oversimplifying, the Arctic Circle theorem says that if you randomly tile a shape called an “Aztec Diamond” with dominoes you’ll almost certainly end up with a really simple pattern near the corners of the diamond.

A picture is worth way more than words here, so here’s a picture of one of the shapes we looked at tonight:

and here’s a description of the idea that is probably a bit better than I could give:

A discussion of the Aztec Diamond / Arctic Circle theorem on Wikipedia

and here’s some software I found that helps you play with tilings of Aztec diamonds:

Dan Romik has software for playing with the Aztec Diamond

So, after hearing about the problem I spent the day thinking that kids would probably really enjoy hearing about / looking at this problem. After finding the software above I couldn’t resist sharing the idea with my kids tonight!

First I had my older son look at the program as we looked at larger and larger tilings. I didn’t explain much of anything about what was going on, I just wanted to hear what he had to say:

Next we went through it again, but I told him a little bit more about what was going on to see

Now I repeated the same process with my younger son – here are his thoughts when he was seeing the tiling patterns for the first time:

and here are his new thoughts after I explained just a little bit more about what he was seeing:

I really think there’s a great project for kids hiding in here somewhere! Maybe it is a computer project, or maybe it is a simple project with snap cubes. It is really amazing to see how order comes from randomness here. Can’t wait to think a little more about how to share the ideas here with kids ðŸ™‚