Revisiting (again!) the Paradox of the Second Ace

Today my younger son and I are going back to a problem that we’ve looked at before – the “Paradox of the Second Ace.” This is a problem I learned about from Futility Closet:

Here’s the problem as described on their site

This problem teaches a couple of good counting lessons. Today we focused on the first part – if you have at least one ace, what is the probability that you have more than one. First, though, we talked through the problem to make sure my son understood it:

Next I asked my son to work through the calculation for the number of hands that have “at least one ace.” He made a pretty common error in that calculation, and we discussed why his calculation wasn’t quite correct:

Now we talked about how to correct the error from the last video via complementary counting:

Now that we had the number of hands that had at least one ace, we wanted to count the number of hands with more than one ace. My son was able to work through this complementary counting problem, which was really nice to see:

Finally, since we had all of our numbers written down as binomial coefficients and these numbers were going to be difficult to compute directly, we went to Mathematica for a final calculation:

Excited to continue this project tomorrow and hear my son’s explanation for the seeming paradox.

Going back to James Tantons’ amazing Möbius Strip cutting project

James Tanton’s Solve This book is full of incredible math projects to do with kids:

Today we went back to do a project that we’ve looked at a few times before – cutting various versions of a Möbius Strip.

We started with a cylinder just to get going with an easy shape:

Next we moved on to a Möbius strip – but when we made this one we didn’t notice that we’d made a full twist rather than a half twist. So we sort of got two surprises. I’ve included this mistake to show that you do have to be a little careful when you do this project – it is easy to accidentally make the wrong shape:

Here’s the correction and the actual Möbius Strip:

Next was a shape where we had a 2nd little error – this time the tape held the final shape together a bit too much. In our video you can sort of guess what the actual shape will be, but I definitely encourage everyone to try this one out for yourself!

The 4th shape is where things really get interesting. For this and the next two shapes we are making a circular cut through a cylinder with a hole cut out of it. Each of the shapes has a interest set of twists in it – the shape in this video has a single half twist on one of the “arms”:

The 5th shape has half twists (going the same direction) in each of the arms:

The 6th shape has half twists going in different directions in each of the arms:

This really is an amazing project for kids. As our version of the project here shows (accidentally!), you do have to be careful with the preparation, but even with the couple of errors we had, it was still an incredibly fun morning.

A geometry puzzle from Catriona Agg that is also great for showing students some ideas from algebra

Last week I saw another terrific puzzle from Catriona Agg:

I left two copies of the puzzle for my son to work through while I was out this morning. For the first run through I asked him to solve the puzzle as it was stated. Here’s his work and his explanation:

For the second run through I asked him to solve the problem assuming that the radius of the circle was X rather than 5. This was first step in what I was hoping would be an interesting algebra exercise. Here he was successfully able to use the quadratic formula even though the equation he found had 2 variables:

For the last part of the project I wanted to see if he could factor the equation he found in the last video. This turned out to be a significantly more difficult challenge, but he figured out how to do it just as we ran out of space on the memory card!

I suspected that the factoring challenge would be more difficult than simply using the quadratic formula, though I didn’t realize how much different it would be. I might try to find some more challenges that involve multiple variables just to get a bit more practice with these ideas.

Why I love sharing ideas from research mathematicians with my kids

Yesterday we did a project inspired by the great podcast conversation between Steven Strogatz and Federico Ardila here:

That project is here:

https://mikesmathpage.wordpress.com/2021/04/03/revisiting-the-permutahedron-with-my-younger-son-after-listening-to-steven-strogatzs-interview-with-federico-ardila/

Last night I asked my younger son what he wanted to do today for a project and he said that he wanted to talk about the permutahedron a bit more. In yesterday’s project we talked about the permutations of the set (1, 2, 3, 4), so today we started by going down to some simpler sets of permutations:

Next we looked at the shape made by the permutations of the set (1, 2, 3). The way my son thinks through this problem shows why I love sharing ideas from math research to my kids.

To wrap up today we dove a little deeper into one of the ideas we talked about yesterday – in the permutations of the set (1, 2, 3, 4) is there a permutation that requires 4 or more flips to get back to the starting point of (1, 2, 3, 4)?

The permutahedron is a really neat shape to explore with kids, and hearing them talk about and think through the shape itself is incredibly fun.

Revisiting the permutahedron with my younger son after listening to Steven Strogatz’s interview with Federico Ardila

This week I listened a great conversation between Steven Strogatz and Federico Ardila on the Joy of X podcast:

We have played with the shape a few times before – see these blog posts of you are interested in seeing other ways that kids can explore the shape:

Today we explored the and idea that Ardila discussed in the podcast – finding paths on the permutahedron.

We started by just reviewing what the shape is and what it represents:

Next we tried an easy example of finding a path on the permutahedron going from a random permutation back to the correct order:

For the last part of the project we tried a more complicated scramble of the cards and found that walking back to the unscrambled state would take a minimum of 3 steps:

I love playing with this shape with kids. It is a great way to get them talking about fairly advanced mathematical ideas and also allows them to see a really neat 3d shape that research mathematicians find interesting!

Working through a neat circle / rectangle puzzle from Catriona Agg

Earlier this week I saw a really neat puzzle from Catriona Agg:

We’ve done lots of projects with Catriona’s puzzles in the past, so just search for “Catriona” and you’ll find them.

My younger son spent some time off camera solving the puzzle and then I asked him to walk through his solution. His solution gets the main idea about tangents and circles, and then computes the radius of the semicircles using the Pythagorean theorem:

Typically when we play with one of Catriona’s puzzles I have my son look through the twitter thread afterwards and find a neat solution. I took a different approach today and showed him how to use similar triangles to get to the answer with slightly less computation:

I really like Catriona’s puzzle. I also think that my son’s explanation is a great example of what kids doing math looks like.

Showing that 1/40 is in the Cantor set is a great arithmetic exercise for kids

Yesterday we did a fun project on the Cantor set inspired by an amazing tweet from Zachary Schutzman:

That project is here:

Today we extended some of the ideas from that project by showing that the number 1/40 is in the Cantor set. Here’s how my son approached the problem – the idea he uses builds on the idea we talked about with the number 1/10 in yesterday’s project. I was happy to see that those ideas had stuck with him!

Now that we knew 1/40 was in the Cantor set, we talked about what other numbers of the same form must be in it. Although we don’t prove it (that’s what the paper in Schutzman’s tweet does), he’s now found all of the numbers with finite decimal expansions that are in the Cantor set

Finally, I wanted to go down a path relating these base 3 expansions to infinite series, but my son’s ideas took this last part in a slightly different direction. Which was fine and also fun. It really shows that kids can have fun exploring – and also have the capacity to have some great ideas about – infinite series.

These two projects have been really fun. I think the ideas about the Cantor set are great for kids to play around with!

A fun fact about the Cantor set and a great arithmetic exercise for kids!

Yesterday I saw an amazing tweet about the Cantor set:

The amazing paper posted by Zachary Schutzman was in response to this question posed by Jordan Ellenberg:

I thought explaining some of the ideas about the Cantor set to my younger son and then having him play around with some fractions in base 3 would make a pretty fun project. So we tried it out tonight.

First we talked a bit about the Cantor set and he shared some initial thoughts:

Next I asked him to try to compute 1/4 in base 3. I always like projects like these with kids as they sneak in a little extra practice with fractions. Here’s his work:

Finally, I asked him to compute 1/10 in base 3 using an idea I mentioned at the end of the last video. After he did that, I asked him to find a few other fractions of the form k/10 that must also be in the Cantor set.

This was definitely a fun project. The math ideas here are slightly tricky, but hopefully the work here shows that the are accessible (and interesting!) to kids.

Talking about the Gaussian Integers with my younger son

My younger son is reading The Book of Numbers by Conway and Guy right now:

Last night he read about the Gaussian and Eisenstein Integers. Today we talked about the them (well, just Guassian integers).

Here’s the introduction to what they are in his words:

Before diving into the Gaussian integers, I asked him to give some of the ideas he thought were important in the regular integers. The talk here gave us some good things to talk about in the next video:

Now we talked about Gaussian integers – one fun thing he talked about here was that multiplication by i was the same as rotating by 90 degrees:

To wrap up today’s project we talked about why the number 5 is not a prime in the Gaussian integers. It was nice that he was able to show this!

I’m excited to introduce the Eisenstein integers tomorrow – hopefully today’s project helps prepare for the slightly more complicated math we’ll see there.

Continuing our exploration of Collatz-like sequences

Yesterday my younger son and I did a fun project inspired by this blog post from Stephen Wolfram:

That project is here:

https://mikesmathpage.wordpress.com/2021/03/06/playing-with-the-collatz-conjecture-inspired-by-a-stephen-wolfram-blog-post/

Today we continued our exploration and started with a quick look at the 5n+1 problem we finished with yesterday:

Next we moved on to the mod 3 version that Stephen Wolfram looks at in his blog post.

Finally, we looked at what happens for the number 469 in this version of the Collatz conjecture, and talked through some ideas about how we would study if it ever repeats.

This was a fun exploration with my younger son. Wolfram’s blog post is incredible, but also accessible to kids. I love being able to explore ideas like these with kids.