This problem gave my son some difficulty yesterday – it is problem #19 from the 2011 AMC 10a

Last night we talked through the problem. The talk took a while, but I was happy to have him slowly see the path to the solution. Here’s his initial look at the problem:

Next we looked at the equation . Solving this equation in integers is a nice lesson in factoring. Unfortunately by working a bit too quickly he goes down a wrong fork for a little bit.

In the last video we found that the original population of the town might have been 484, and it might have changed to 634 and then once more to 784. We had to check if 784 was a perfect square.

Finally, we needed to compute the approximate value (as a percent) of 300 / 484. The final step in this problem is a great exercise in estimating.

So, a really challenging problem, but also a great problem to learn from. We went through it one more time this morning just to make sure that some of the lessons had sunk in.

This week’s video is about philosophy and math. A deep subject, for sure, but one which the kids thought was interesting. Here’s the video (and the twitter link so you know when the new videos appear!):

Need a Thanksgiving dinner topic? Argue about the philosophy of mathematics with the help of our new episode: https://t.co/BjKNgV4RGu

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:

After playing the game for a just a few minutes I knew that my kids would love it.

Here’s each of their reaction to seeing and playing the game.

My younger son first:

My older son next:

So, definitely a fun little game for kids. They need to be fairly fluent with the arrow keys on the keyboard, but that’s really all that’s required. Definitely some fun puzzles to solve!

I happened to see the Raspberry Pi set below at a store earlier in the week:

Â Today I showed it to the kids and we played with it for a bit. Here’s their initial reaction:

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There wasn’t much to assemble which was nice. We did have some trouble connecting it to my laptop, so we switched to connecting it to our TV. Here’s what the boys had to say after we got it running:

Finally, the kids discovered a few of the games that came loaded on the computer. This squirrel one made them laugh. From start to finish was about an hour – and at least 20 minutes of that was trying and failing to get connected to the laptop.

We’ve only scratched the surface of what was in the kit. I’m excited to have the boys play around with the computer a bit more. It has Mathematica (yay!) and some software for introductory programming like Scratch. Hopefully there will be many more projects to come.

Had a great night with the boys tonight. My older son was working on some old AMC 10 problems and we talked through one that stumped him for his movie:

It was #15 from the 2013 AMC 10a:

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Next I spent some time with my younger son. He’s been studying the basics of lines using Art of Problem Solving’s Introduction to Algebra book plus a little bit of Khan Academy (when I’ve been traveling for work). I asked him what he’d learned so far and loved his response. It was a great reminder of the joy of learning new math ideas for the first time:

Everyone always talks about thinking through problem carefully – here’s a great problem and a great opportunity to give some examples of that kind of deep thinking.

Anyway, I ran out to home depot and got some wire and we made some knots. I had each of the boys make a trefoil knot and then make a random knot of their own choosing. In retrospect I wish I’d spent maybe just 5 minutes explaining some of the ideas in Richeson’s blog post – oh well, the excitement got the better of me ðŸ™‚

Here’s my older son playing with his trefoil knot and making a Mobius strip bubble. I love the “hey, I actually think I got it” moment:

Here’s him playing with the knot me made – in retrospect I’d argue for a knot that was slightly less complicated:

Next up was my younger son. First up was the trefoil knot and we got another great moment “I think this might be a Mobius strip” !!

Finally we made his own knot and explored. Again, I’d probably ask for a less complicated knot if I was doing this again:

So, that so much to Dave Richeson for posting his old project – this is an incredible project, and an especially great one for kids. The appearance of the Mobius strip is really quite an amazing little math miracle!

Here’s what my kids thought was neat about the video:

My younger son thought the “central sphere” problem was fun, and my older son thought the shape of the n-dimensional spheres was fun. I originally intended to talk about both this morning, but our talk about the shape of the spheres took enough time for one project.

So, below is our initial look at the shape of the spheres. There’s a lot of nice introductory geometry (and fractions!) in the discussion. Also, I made the choice to talk about 1/2 the length of the long diagonal – that choice sort of confused the kids, so I’d focus on the full length if I was doing this again.

Finally, we talked about how the diagonal changes as you go up in dimension. This is a fairly straightforward application of the Pythagorean theorem, so it isn’t that hard to talk about. The boys saw the pattern fairly quickly.

Then I introduced the volume formula for even-dimensional spheres and we calculated the ratio of the volume of a 30-dimensional sphere to the 30-dimensional box it is inscribed in.

I’m super excited for this new series of videos from Kelsey Houston-Edwards, and I can’t wait to share the next one with my kids!

I was super excited to see this new work from Kelsey Houston-Edwards since, for one(!), I was really hoping that mathematicians would publicize the sphere packing result and find ways to make it accessible to the general public:

I was fortunate to be able to attend Maryna Viazovska’s talk at Harvard and Henry Cohn’s talk about BU about the new sphere packing results. Although I don’t have nearly the mathematical sophistication to be able to write about the result in any detail, both talks were great. There’s also a link in the post below to a nice talk from Henry Cohn about the history of the sphere packing problem, but I think that’s as close as you can get to the problem without diving into very heavy math:

I called the problem “Bjorn Poonen’s n-dimensional sphere problem” because I learned about it from him. So, to be 100% clear, he wasn’t taking credit for the problem. I learned later from Alexander Bogomolny that at least the two parts of the problem discussed in the video were attributed to Leo Moser