# The paradox of the 2nd ace part 2

Yesterday we did a project on this fun problem from Futility Closet:

Today we finished the project by talking about the 2nd part of the problem and then having a discussion about why the answers to the two questions were different. Unfortunately there were two camera goofs by me filming this project – forgetting to zoom out in part 1 and running out of memory in part 4 – but if you go through all 4 videos you’ll still get the main idea.

Here’s the introduction to the problem and my son’s solution to the 2nd part of the problem. Again, sorry for the poor camera work.

Next we went to the computer to verify that the calculations were correct – happily, we agreed with the answer given by Futility Closet.

In the last video my son was struggling to see why the answers to the two questions were so different. I’d written two simulations to show the difference. In this part we talked about the difference, but he was still confused.

Here we try to finish the conversation about the difference, and we did get most of the way to the end. Probably just needed 30 extra seconds of recording time ðŸ˜¦ But, at least my son was able to see why the answers to the two questions are different and the outputs from the simulations finally made sense to him.

So, not the best project from the technical side, but still a fun problem and a really interesting idea to talk through with kids.

# 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.

# Playing with the Collatz conjecture inspired by a Stephen Wolfram blog post

I saw a really neat blog post from Stephen Wolfram last week:

The blog post has so many different ideas that you could share with kids, but I decided to spend the weekend exploring various versions of Collatz-like sequences with my son.

We started by looking at some simple code in Mathematica to generate Collatz sequences:

Next we looked at how long it took various numbers to get to 1 in the Collatz sequence and looked at a histogram of the numbers. We got a fun surprise:

We wrapped up today’s project by looking at what happens when you replace the 3x + 1 rule in the Collatz conjecture with a 5x + 1 rule. I don’t remember ever seeing this idea before and it was one that really surprised me reading through Wolfram’s blog post this morning:

Tomorrow we’ll extend today’s project by looking at the 4x + 2 and 7x + 1 version that Stephen Wolfram mentioned in his tweet. Hopefully that’ll make for a really fun project, too.

# Having the kids play with the Lightbot, a \$3 intro programming app

This week I learned about Lightbot from one of Michael Pershan’s tweets:

Although he was using the app with his 6 year old, I thought it would be fun to see what my kids (in 9th and 11th grades) would think of it. They really enjoyed it, and both played with it for longer than I asked them to.

My older son went first – he has a decent amount of programming experience and is taking a programming class at his high school this year. Here’s what he had to say:

My younger son went next. He as a tiny bit of programming experience. You can hear that the language he uses isn’t as sophisticated as the language my old son used, but that’s fine. You can also see that the app is very easy to learn how to use as this video shows him solving a level with only about 20 to 30 min of playing around with the game.

I was really happy that the kids liked this app as much as they did. Hopefully they’ll play around with it a bit more – it certainly looks like a neat introductory programming game for kids of all ages!

# Revisiting a neat coin flipping game I learned from Ole Peters

This morning I accidentally stumbled on an old coin flipping game we looked at last year:

I thought it would be fun to take a look at the problem again since the last look was long enough ago that the boys probably wouldn’t remember it. Here are their initial thoughts in the problem. After a bit of discussion, the boys came up with a good argument for why HHHT only would appear more often than HHHH only.

Next we looked through a simple computer program I wrote to model the situation. This isn’t the best or most clever way to write the program, but I thought it was an easy one to explain:

Finally, we looked at how the numbers would change if the sequence had 50 flips instead of 20. It was interesting to hear the boys explain why the numbers had changed – I think this extra discussion helped them understand the original problem a bit better:

# Extending our project exploring the PÃ³lya Urn

Yesterday my younger son and I looked at the PÃ³lya Urn problem inspired by this sequence of tweets from Ole Peters and Marcos Carreira:

Today I had my son explore a little further. He was interested to see if different starting positions led to different distributions of endings. He looked at five different starting positions. Here’s the first (with a quick review of the problem) when the urn starts with 5 black and 5 white balls and we play the game 1,000 times:

Next he looked at how the starting position with 1 black ball and 5 white balls evolved. The way the distribution of the number of white balls at the end changes is pretty amazing:

Now for the most surprising one of all – the starting position with 1 white ball and 1 black ball – it seems that ending with 1 white ball or 1001 white balls (or any amount in between!) is equally likely:

Finally he looked at the starting position with 1 black ball and 10 white balls. This one is a little less surprsing having already seen the 1 black ball and 5 white ball game, but still it was neat to see:

This is a fun little game for kids to study. It is also a nice introductory programming exercise, too. Thanks so much to Ole and Marcos for sharing their ideas!

# Playing around with Polya’s Urn thanks to Ole Peters and Marcos Carreira

With the results and the code presented so nicely, diving into Polya’s Urn with my younger so this weekend was a no brainer!

I started by explaining the problem and asking for his thoughts. His intuition about how the game and how it would play out was fascinating to hear:

After he explained what he thought would happen, we played the game once:

Next we went to the computer to explore the game in more detail. Before diving in, though, I had my son explain Carreira’s code:

Finally, we played the game a few times and looked at the different outcomes. The results really are amazing. Hopefully we’ll do a follow up project tomorrow:

# A nice introductory statistics conversation with my younger son

My younger son wants to learn more about statistics. I’m excited to come up with some projects, though our journey here is probably not going to look like a typical statistics class.

My first idea was to have him investigate coin flips and look at similarities and differences with 10, 100, and 1000 coin flips. But really before we even got started he had a really interesting question.

So, here’s how he described the program I had him write ahead of time, and then we discuss the question he had -> In 100 flips, why did it seem that the chance of getting less than or equal to 50 heads was more than 50%?

To start diving in to his question, we first looked at sequences of 100 flips to get a better sense of what was going on. Interestingly, on a test look at 10 sequences, we did find that 6/10 had less than 50 heads:

Now we looked at the distribution of heads in 100 flips using Mathematica’s Histogram function – it was really interesting to hear him describe the different distributions that we saw:

Finally we talked about why my son was seeing what he was seeing in his coin flip program.

# Having my younger son explore a fantastic website shared by Bill Hanage explaining how social distancing changes network connections

Yesterday I saw a tweet from Bill Hanage linking to a really interesting website:

From just a little bit of time on the site, I thought having my younger son read and play with some of the ideas would make a great project. So I asked him to spend 20 min reading and exploring, and then we talked.

Here are his initial thoughts:

One of the things he thought was interesting was the idea of 3 and 6 degrees of separation when you have a few connections and how much the network changes when you just add one connection (on average) per person:

Another thing he thought was interesting was the companion site that allowed you to modify connections in the network. Here he looked at the size of the largest group when you made the change from connections with only essential works to again adding 1 connection on average for everyone:

I really like how the ideas of network connections are explained on this site. Their work makes a fairly complex idea accessible to everyone – including kids. Thanks to Bill Hanage for sharing this site!

# Sharing Grant Sanderson’s Hamming Code video with my younger son

Yesterday Grant Sanderson published a fantastic set of videos on Hamming codes. I watched the first one with my younger son last night:

Today we talked about some of the ideas in the video – starting with some of the things he thought were interesting:

Next I had him work through one of the examples in Grant’s video – I didn’t realize it was an example of an error since I just pulled it off of a screen shot, but we discovered the error talking through the example:

Finally, we went back to the same example. This might seem like a strange thing to do, but Grant’s example had an error in the parity bit and I wanted to make sure my son understood that the error correcting codes could also detect that kind of error.

I love Grant’s work – it makes for such a fun and easy way to explore ideas that kids wouldn’t normally see in their school math.