## A simplified version of the Banach-Tarski paradox for kids

Yesterday we were listening to Patrick Honner’s appearance on the My Favorite Theorem podcast. Honner was discussing Varignon’s Theorem. We actually have discussed this appearance before, but the kids hadn’t listened to the podcast, yet:

After listening to the podcast I asked my older son what his favorite theorem was:

However, after giving up on the idea initially (!) I looked at the Wikipedia page for the Banach-Tarski paradox and found an idea that I thought might work. Here’s the page:

The idea was to share the first step in the proof – exploring the Cayley graph of $F_2$ – with kids. Here’s the picture from Wikipedia:

So, here’s what I did.

First I introduced the boys to some basic ideas about a free group on two generators. I used a Rubik’s cube to both demonstrate the ideas and to show why a Rubik’s cube didn’t quite work for a perfect demonstration (I know that part of the video drags on a bit, but stay with it – there is a nice surprise):

Next we talked about the free group with two generators in more detail. My younger son accidentally came up with a fantastic example that helped clarify how this free group worked.

Then there was a bit of a surprise misconception that I only uncovered by accident. That led to another important clarification.

So, completely by accident, we had a great conversation here.

In the last video they boys thought you could use the “letters” $x, x^{-1}, y, y^{-1}$ only once. In the beginning of this video I clarified the rules.

Next we began to talk about the representation of our free group by the Cayley graph from Wikipedia pictured above. I was really fun to hear how the boys described what they saw in this graph.

Finally, we looked at two different ways to break the Cayley graph into pieces. This video is a little long, but it has a simplified version of the main idea in the Banach-Tarski paradox.

The first decomposition of the Cayley graph is into 5 pieces -> the identity element, words that start with $a$, words that start with $a^{-1}$, words that start with $b$, and words that start with $b^{-1}$. This decomposition is pretty easy to see in the picture.

The second – and very surprising decomposition is as follows:

The combination of (i) the words that start with $a$ and (ii) $a$ multiplying (on the left) all the words that start with $a^{-1}$ gives the entire set. The same is true for the combination of (i) the words that start with $b$ and (ii) $b$ multiplying (on the left) all the words that start with $b^{-1}$

Although the words describing this decomposition might not make sense right away, you’ll see that the boys had a few questions about what was going on and eventually were able to see how this second decomposition worked.

And this second decomposition gives a huge surprise -> we’ve taken 4 subsets, combined them in pairs and created two exact copies of the original set. Ta da 🙂

This project is an incredibly fun one to share with kids. I’m pretty surprised that *any* ideas related to the Banach-Tarski paradox are accessible to kids, but the simple ideas about the Cayley graph of $F_2$ really are. Using those ideas you can show the main idea behind the sphere paradox without having to dive all the way into rotation groups which I think are a little more abstract and harder to understand.

Anyway, this one was a blast!

## Another great problem from Matt Enlow’s collection

Today we talked about another problem from the amazing list of problems that Matt Enlow’s published a few weeks ago:

This is our second project from that collection. The first is here:

Sharing a neat problem from Matt Enlow with kids

For today’s problem I introduced the problem and asked the boys for their initial thoughts. My older son noticed an important property about the sum of 9 and consecutive integers. He explained the property that the sum of 11 consecutive numbers would have and then my younger son explained the similar property that the sum of 9 consecutive numbers would have:

Next we had to see if there were any special properties that the sum of 10 consecutive integers would have.

Once we had that property, my younger son was able to explain how you could use them to find a number that would work (though not necessarily the smallest one):

At the end of the last video we though that 495 would satisfy the conditions of the problem. Here we checked that it did and wondered if it was the smallest.

Finally, we checked to see if 495 was indeed the smallest positive number with the properties in the problem.

My older son thought that 0 would have worked, but working it out he saw that it didn’t.

After that, we saw that 495 was indeed the smallest.

Definitely a great problem – it is nice to hear the boys explain some basic ideas in number theory. It is also a nice problem because kids – well, at least my kids – often struggle to see the difference between “find the smallest” and “find an example” and this problem helps show that “find the smallest” requires a bit more work.

## A review of the game Pylos

Thanks to a tweet from Vincent Pantaloni last week I learned about the game Pylos:

This morning my younger son finished up his math work a little early so we decided to do a quick review of the game.

Here’s the game and my son’s thoughts about it after having played one game last night – I love that he sees that one of the math ideas involved in this game is sphere packing!

Next we demonstrated a game to show that the play itself is pretty easy for kids, yet filled with interesting strategy:

We wrapped up by discussing the game. I think my son’s comment that the game is surprisingly simple and fun is a great summary. Definitely a great game for kids.

## Happy Pi Day

For Pi day today we explored the amazing near integer $e^{ \pi \sqrt{163}}$

I started by showing the boys the numbers as well as just how close it was to being an integer. I measured the closeness both in terms of the decimal expansion and in terms of the continued fraction expansion of the number:

Next I asked the boys to each take a turn finding another number relating to $\pi$ that was either nearly an integer or nearly a rational number. It turned out – especially with my younger son – to be a really nice way to discuss properties of powers of numbers.

The number my younger son found was $100 * 314^{\pi}$

The number my older son found was $3.4 * \pi^{\sqrt{2}}$

So – obviously just for fun – but still a neat way to talk about numbers and continued fractions. And a pretty fun number at the start, too 🙂

## A review of “The Sherlock”

We received a nice gift from Jim Propp earlier this month. With the kids off of school for a snow day today, it seemed like a good time to open it up.

My younger son played with it for a while and then I wanted to have him show how the game worked. Here are his initial thoughts about the game:

Here’s the first example of a puzzle solve. You’ll see that even having solved it once before it is still not necessarily so simple:

Here’s a second solve example – this one goes pretty quickly

Finally, we wrapped up by having him show some of the other pieces and me asking him to talk about what he thinks the main ideas are for this puzzle. Interestingly he doesn’t think that it is a math puzzle, but rather a logic puzzle 🙂

So, thanks to Jim Propp for giving us this really nice puzzle game.

## Sharing Stewart’s theorem with my son

My older son had a problem about finding the length of an angle bisector in a 3-4-5 triangle in his enrichment math class last week. Solving this problem is a little tedious, but also gives a great opportunity to introduce Stewart’s theorem. I first learned about Stewart’s theorem from Geometry Revisited when I was in high school. Here’s an explanation of the theorem on Wikipedia:

Stewart’s theorem on Wikipedia

I started off the project tonight by reviewing the original problem with my son:

Next I briefly introduced the theorem and then we got interrupted by someone knocking on our front door:

Now I showed how the proof goes. We had a brief discussion / reminder about the relationship between $\cos(\theta)$ and $\cos( 180 - \theta )$ and after that the proof went pretty quickly:

Finally, we returned to our original triangle to compute the length of the angle bisector using Stewart’s Theorem. The computation is still a little long, but now the calculations themselves are pretty straightforward:

Definitely a beautiful theorem. It is amazing that the law of cosines simplifies so nicely and that computing the lengths of cevians of a triangle.

## Sharing a neat problem from Matt Enlow with kids

Yesterday Matt Enlow shared a list of his 100 favorite problems:

I flipped through the problems yesterday and problem #6 struck me as a terrific one to share with kids:

I mentioned the problem to the boys yesterday and today we dove into it. Here are their initial thoughts:

Next I asked them to see if they could say anything at all about what would have to be true if there were powers of 2 and 3 that met the conditions of the problem.

My older son noticed a pattern in the powers of 2 mod 3. That helped us understand some basic ideas about what would have to be true if powers of 2 and 3 differed by 1. We then moved on from that idea to see how the “difference of squares” idea from algebra could help us show that the equation in the problem would probably never be true for an even power of 2 greater than 4. Nice start – now we just had to get to the finish line:

The idea that we were missing at the end of the last video was that powers of 3 only had 3 as a prime factor. Once the boys noticed that, they were able to see that an even power of 2 could never satisfy the equation!

Now we had to look at odd powers of 2. They noticed that roughly the same idea works if the power of 3 was even. There was one little subtle difference in the argument, but luckily both boys were able to explain that bit!

Now we had to look at the case with odd powers of 3 and odd powers of 2. Here I showed them how polynomials like $x^n - 1$ factor. I also shows how the numbers of the form $3^n - 1$ factor when n is odd.

The interesting idea here was that the factorization was always a 2 and an odd number. That showed the product could never be a power of 2. It took a while for us to get to that via the polynomial factoring, but we did get there.

Which then solved the whole problem!

Finally – just to wrap things up, I went to the computer to find powers of 2 and 3 that were “close” together using continued fractions:

I was lucky to see Matt Enlow’s list of problems on twitter yesterday. It is going to be a great resource for me – can’t wait to share more of the problems with the boys.

## Comparing a tetrahedron and a pyramid with theory and experiment

We’ve done a few projects on pyramids and tetrahedrons recently thanks to ideas from Alexander Bogomolny and Patrick Honner. Those projects are collected here:

Studying Tetrahedrons and Pyrmaids

One bit that remained open from the prior projects was sort of a visual curiosity. When you hold the zome Tetrahedron and zome Pyramid in your hand, it doesn’t look at all like the pyramid has twice the volume. Today’s project was an attempt to dive in a bit more into this puzzle.

We started by reviewing the ideas that Alexander Bogomolny and Patrick Honner shared:

Next we reviewed the geometric ideas that lead you to the fact that the volume of the square pyramid is double the volume of the tetrahedron.

Now we moved to the experiment phase – we put packing tape around the tetrahedron and the pyramid and filled them with water (as best we could). We then dumped that water into a bowl and used a scale to measure the amount of water. Our initial experiment led us to conclude that there was roughly 1.8 times as much water in the pyramid.

After that we repeated each of the measurements to get a total of 5 measurements of the volume of water in each of the shapes. Here are the results:

Definitely a fun project. I wish that we’d have gotten measurements that were closer to the correct volume relationship, but it is always nice to see that experiments don’t always match the theory!

## Sharring problem A1 from the 2017 Putnam with kids

We had a snow day today and I finally got around to sharing a neat problem from the 2017 Putnam Exam with the boys.

When I first saw the problem I thought it would be absolutely terrific to share with kids:

I started off the project today by having them read the problem and spending a little bit of time playing around:

After the initial conversation the boys, I triehd to start getting a bit more precise. The first sequence of numbers they knew was in the set was 2, 7, 12, 17, . . . .

They were not sure if 4 was in the set or not. My first challenge problem to them was to show that if 4 was in the set, then 3 would be in the set.

My next challenge question was whether or not 1 would be in the set.

Now we moved on to one of the number theory aspects of the problem – is 5 in the set?

During this conversation my younger son noticed that we had found a number that was 1 mod 5.

Finally, we talked through how you could find 6 from the number my son noticed in the last video.

I’m really happy with how this project went. This problem is not one (obviously) that I would expect the kids to be able to solve on their own, but most of the steps necessary to solve the problem are accessible to kids. It was really neat to hear their ideas along the way.

## Sharing “developable surfaces” with kids thanks to a brilliant lecture from Heather Macbeth

[This is a redo of a blog post from January 2018 that somehow ended up 1/2 deleted. Not sure what I did to that old post, but I didn’t want to lose the ideas.]

In January 2018 I attended a terrific public lecture given by Heather Macbeth at MIT. The general topic was differential geometry, and the specific topic she discussed was “developable surfaces.”

Here’s an example from the talk:

I also printed a few examples and shared them with the boys the next day: