# Federico Ardila’s “Combinatorics and Higher Dimensions” video is incredible

Yesterday Numberphile published an absolutely amazing new video with Federico Ardila:

The video blew me away – it felt like such a great way to share ideas about higher dimensions with kids.

This morning I shared the video with my older son (my younger son had some school homework that he forgot to do . . . ). Here are his thoughts after seeing the video. One surprise (to me) is that he thought this way of thinking about subsets only works for small sets.

Next I had him label the corners of a cube using the method that Ardila discusses in the Numberphile video. Sorry that the labels on the tape didn’t show up so well, but I think the idea still comes through:

Next – we talked about two versions of the hypercube that we’ve looked at in the past. Then I asked him to pick one of those hypercubes and label the verticies using the ideas from the Numberphile video. I really believe that Ardila’s idea lets kids experience a 4 dimensional cube in a completely new way:

If you know kids who are interested in understanding the 4th dimension, have them watch and they play around with Ardila’s video. It is completely amazing!

The “standing a cube on a corner” ideas we talked about in the 2nd and 3rd videos come from this video from Kelsey Houston-Edwards. Here’s her video and the two projects that we did from it – using the ideas from her video in combination with the ideas in Ardila’s video give kids an amazing look at the 4th dimension

Kelsey Houston-Edwards’s hypercube video is incredible

One more look at the hypercube

# Exploring a 13-14-15 triangle

My younger son was struggling with one of the challenge problems from Art of Problem Solving’s Introduction to Geometry book today. I hadn’t done a project with him this week, so I thought it would be fun to turn talk about this problem in detail.

Here’s the introduction to the problem and some of his initial thoughts:

He had a pretty good idea about how to proceed by the end of the last video. Here we started to work through the algebra. There is one simplification that helps get to the solution fairly quickly – it took him a little while to find it. Once he did, though, the solution came quickly.

Now all that was left was a little arithmetic and checking that the answer we found was correct.

At the end he gives a nice summary of the problem.

It turns out that I did a previous project on the 13-14-15 triangle with my older son:

Mr. Honner’s 13-14-15 triangle and a surprising unsolved problem

Amazing how useful this triangle is!

# A terrific problem to share with calculus and geometry students from the Iowa State problem collection

This is another problem from the Iowa State problem collection:

Yesterday I looked at one of the number theory problems with boys:

Some of the Iowa State “problems of the week” are great to share with kids

Tonight I tried another terrific (though very challenging) problem with my older son:

Here’s are his initial thoughts about the problem:

Now we rolled up our sleeves a bit and started to solve the problem. His first thought about what to do was to try to solve the problem with one inscribed circle and then with three inscribed circles:

The problem with three inscribed circles was giving him trouble so we moved on to a new movie and sort of started over on the three circle problem. While he was re-drawing the picture he was able to see how to make some progress:

Finally, having solved the problem with three circles, he moved on to solving the problem in general and found the surprising answer:

I really like these problems. Obviously not all of them are going to be accessible to kids, but the ones that are accessible are really amazing treasures!

# Some of the Iowa State “problems of the week” are great to share with kids

Saw a tweet about a really nice collection of problems yesterday:

As I clicked through a few of the problems last night, I thought that several (though definitely not all!) would be nice ones to share with kids. When my older son got home from school today I asked him about the problem from January 8th, 2018:

Here’s what he had to say – it is a really nice solution:

Next we moved on to the 2nd problem from January 2018:

He walked away before we could start solving this one . . . . ðŸ™‚

Update:

When my younger son got home from school I asked him the same question. His work shows what a kid thinking through a math problem can look like:

# One that didn’t go so well – talking about knight’s tour problems with the boys

Saw a neat tweet from Joel David Hamkins at the end of last week:

I thought it would be fun to talk through the knight’s tour problems with the boys today and end by showing them the infinite problem. I ran into trouble almost immediately when we began to talk about the tours on the 3×3 and 4×4 boards. The difficulty they had explaining was a big surprise to me. We ended up talking about the 4×4 problem for almost 30 min.

Tonight I sat down with each of them and asked them to talk me through the problem and explain why the knight’s tour on the 4×4 board was impossible. You can see that my older son (in 8th grade) was able to explain the problem pretty well, but my younger son (in 6th grade) still really struggled.

Here’s what my older son had to say:

Here’s what my younger son had to say:

Definitely a much harder problem for kids than I thought. Hopefully will have some time during the week to explore this and maybe a few other tour problems with them.