# Having the boys try out some fun and surprising ideas about 360 and 720 degree rotations from Eric Weinstein’s interview with Roger Penrose

Yesterday I heard a terrific interview with Roger Penrose on Eric Weinstein’s podcast:

The podcast episode is here:

‘Today the boys and I talked through three fun to see, but maybe tricky to understand, ways that something need to rotate 720 degrees to get back to where it started.

We started by looking at a circle rotating around a second circle of the same size:

Next we looked at the famous “wine glass” problem. I originally wanted to color the water in the glass with food coloring, but chickened out!

Before going on to the Dirac Belt Trick, I showed the boys this really nice video showing the trick in a pretty unusual – and super fun – way:

After the video demonstration, I had the boys try the trick with a belt. At the end my old son made a connection between the belt trick and the complex numbers which was a nice and totally out of the blue surprise to me:

Anyone interested in physics should listen to Weinstein’s interview of Penrose – it is amazing. I was really happy to be able to pull out a few ideas from the interview to share with the kids today!

# Introducing the game of Hex to my younger son

I recently got a book about the game of Hex:

I’m hoping to be able to use some ideas form the book for fun projects with the boys. Today I thought it would be fun to introduce the game and try out a challenge problem from the book. It was an interesting project – the game was harder for my son to think about than I was expecting, but I have some new ideas to try out now.

Here’s how today went -> first an introduction and a quick game:

Now we talked a bit about what he thought some good strategies would be and we played again:

Next we tried a challenge problem – it was pretty difficult, but gave rise to some really good discussion. Here’s the first 5 min of thinking about it:

Finally, here’s the rest of the discussion on the challenge problem – I had to give the answer, but we did play it out from there:

I think playing around with this game is going to be fun. I definitely didn’t gauge the difficulty of the challenge problem correctly. Hopefully I’ll figure out how to use the problems from the book a bit better in future projects.

# A “new to me” demonstration of the difference between Nassim Taleb’s “mediocristan” and “extremistan” thanks to Steve Phelps

Yesterday I saw a really neat tweet from Steve Phelps:

The idea he is studying goes like this:

Select three points uniformly at random inside of a unit square. What is the expected area of the circle passing through those three points?

This question turns out to have a lot of nice surprises. The first is that exploring the idea of how to find the circle is a great project for kids. The second is that the distribution of circle areas is fascinating.

I started the project today by having the kids explore how to find the inscribed and circumscribed circles of a triangle using paper folding techniques.

My younger son went first showing how to find the incircle:

My olde son went next showing how to find the circumcircle:

With that introduction we went to the whiteboard to talk through the problem that Steve Phelps shared yesterday. I asked the boys to give me their guess about the average area of the circle passing through three random points in the unit square. Their guesses – and reasoning – were really interesting:

Now that we’d talked through some of the introductory ideas in the problem, we talked about how to find the area of a circle passing through three specific points. The fun surprise here is that finding this circle isn’t as hard as it seems initially:

Following the sketch of how to find the circle in the last video, I thought I’d show them a way to find the area of this circle using ideas from coordinate geometry and linear algebra – topics that my younger son and older son have been studying recently. Not everything came to mind right way for the boys, but that’s fine – I wasn’t trying to put them on the spot, but just show them how ideas they are learning about now come into play on this problem:

Finally, we went to the computer to look at the some simulations. The kids noticed almost immediately that the mean of the results was heavily influenced by the maximum area – that’s exactly the idea of “extremistan” that Nassim Taleb talks about!

This project is a great way for kids to explore a statistical sampling problem that doesn’t obey the central limit theorem!

I really love the problem that Phelps posted! It is such a great way to combine fascinating and fundamental ideas from geometry and statistics

# An introductory mathematical induction problem I learned thanks to Kelsey Houston-Edwards

Yesterday Annie Perkins asked about resources to help a student understand mathematical induction:

Looking for some resources I was reminded of this incredible video that Kelsey Houston-Edwards made in 2017 – the proof by induction example starts around 5:25:

Today I thought it would be fun to have the boys go back and work through that problem. I started by introducing the problem after we’d made some L shapes out of snap cubes. The boys were able to understand the problem and solve the 1×1 and 2×2 cases:

Now we moved to the 4×4 case. My older son started by looking at different cases – which was a good way to solve the specific 4×4 case. At first it was hard to see how to generalize this approach, though my younger son had some nice ideas about how to extend it:

The next step was a deep dive into the 4×4 case and looked carefully at the exact problem we were trying to solve. Here the boys did a great job of seeing how to extend the solution of the 2×2 case to the 4×4 case.

Finally, we looked at how to extend the 4×4 case to the 8×8 case, and then how to extend to all cases of the problem. We ended the project by discussing how / why mathematical induction works on problems like this one:

# Finding the coordinates of the vertices of a Tetrahedron and an Octahedron

My younger son is starting to learn about coordinates in 3 dimensions. I thought that spending a little time finding the coordinates of the corners of a tetrahedron and an octahedron would make for a nice project this morning.

We started with the tetrahedron and found the coordinates for the bottom face. Once nice thing about the discussion here was talking about the various choices we had for how to look at the tetrahedron:

Having found the coordinates for the bottom face, we now moved on to finding the coordinates for the top vertex:

Now we moved on to trying to find the coordinates for the corners of the octahedron. Here the choices for how to orient the object are a little more difficult:

Finally, we talked through how we would find the coordinates of the octahedron if we had it oriented in a different way. This was a good discussion, but was also something that confused the boys a bit more than I thought. We spent about 10 min after the project talking through how to find the height. Hopefully the discussion here helps show why this problem is a pretty difficult one for kids:

# Playing with Pascal’s triangle and angles hidden in cubes

In April 2018 I saw a great Numberphile video with Federico Ardila:

The project that we did after seeing that video is here:

Federico Ardila’s Combinatorics aand Higher Dimensions video is incredible

This week my younger son was learning about coordinates in 3 dimensions in his precalculus book and I though it would be fun to revisit some of the ideas about cubes from Ardila’s video.

We started by looking at cubes in 0 to 4 dimensions and discussing how we could see Pascal’s triangle hiding in the cubes:

In the last video we got a little hung up on the 4-dimensonal cube, so for the next part of the project we looked at the coordinates of the vertices of the various cubes to see if that could help us see Pascal’s triangle in the 4d cube.

Next we moved on to looking at the angles made by the long diagonal in the various cubes. This exercise was particularly nice since my younger son has been learning a little trig and my older son has been learning a bit of linear algebra.

For the final part of the project we looked at the 4-d cube. Here are zometool shape isn’t really helping us see the long diagonal. My younger son did a really great job seeing the pattern in the right triangles with the long diagonal. He also noticed the amazing fact that there is a 30-60-90 triangle hiding in a hypercube!

My older son was also able to find the same angle using ideas from linear algebra:

Definitely a fun project. It is fun to introduce coordinates not just for 3 dimensions, but for all dimensions at the same time. There’s also an enormous amount of fun math hiding in the seemingly simple idea of n-dimensional cubes, which makes this project sort of doubly fun!

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# Sharing an interesting (and famous) population sampling problem with kids

I saw a great thread on twitter last week – actually in the reverse order in which the tweets appeared. First I saw from Vincent Pantaloni:

Which led me to this amazing tweet from Andrew Webb:

I thought it would be fun to do a project on this idea with the boys. Unlike a few (or maybe most) of our introductory statistics exercises, the program here was likely going to be too hard for the kids to write themselves, so I just wrote it myself and the boys played with it at the end.

To start I had them look at Pantaloni’s tweet:

Next we looked at Webb’s tweet – this one requires a bit more explanation, but the boys were able to understand what Webb’s animation was showing:

Now I spent 5 min explaining how the program I wrote worked. Since my simulation was quite a bit more simplified than the prior two (and also didn’t have any animation), I wanted to be sure they understood what I was doing before we dove in:

For the first run of my simulation, we looked at a 5000 trials of a pond with 1,500 fish and sampling from 4% of the pond From the conversation here you can hear that the boys are gaining a pretty good understanding of the process and are also able to make sense of the distribution of outcomes:

Finally, we looked at 5000 trials of a pond with 750 fish and sampling from 16% of the pond. Again the boys did a nice job explaining the results.

At the end we talked about why this sort of sampling problem can be really difficult.

# Exploring some probability ideas from poker played with 2 decks of cards with kids

On New Year’s eve my older son was playing poker with his friends using 2 decks of cards – see if you can spot the oddity in the picture 🙂

For today’s project we decided to explore some of the probability ideas around playing poker with 2 decks of cards. First we just looked at the possible hands and talked about some potential questions to ask:

For an introductory problem, we looked at the number of ways of getting a Royal Flush and then all types of flushes with a 5 card hand dealt from a single deck of cards:

Now we looked at how regular flushes could happen when dealt from a two decks of cards shuffled together. We also had a good discussion about whether or not it was more likely or less likely to get a flush in the 2 deck situation:

Finally, we went back to Mathematica to take a look at the numbers for a general flush with one and two decks. Here we are lumping all kings of flushes together – regular ones, straight flushes, and Royal flushes are all the same.

Compute the chances of various hands with 2-deck poker is a pretty fun math exercise for kids.