Last week I attended a summer lecture at MIT by Aaron Pixton about a curious math idea. Here’s the short description of the lecture:

“Although the series 1 + 2 + 3 + … diverges, zeta function regularization gives this sum the curious value of -1/12. I will discuss two ways of making sense of summing similar divergent series such as … + -2 + -1 + 0 + 1 + 2 + … that are doubly infinite rather than singly infinite. I will then explain how to generalize to sums over lattices of higher rank, as well as how to interpret parts of this as giving a summation formula for certain finite sums.”

The details from the lecture are too advanced for my kids (and probably for me, to be honest!) but I still thought it would be fun to revisit the old video from Numberphile about the series.

Since we’ve talked about the series before, I thought the best way to get started today was to get some thoughts from the boys about this strange result.

Their ideas are fascinating – I was surprised that both of them seem to have come around to believing the result. The first time we viewed the Numberphile video (when it first came out) they were very skeptical:

Next we watched the video again – so here it is:

Having watched the video, I asked they’ve changed their mind or had any extra suspicions. Again, to my surprised they were now even more convinced.

They did think a few things were strange, though, so at least they were thinking carefully about the result.

Next we tried to recreate the “proof” from Numberphile’s video. This section went pretty well.

Finally, I mentioned some of the ideas from the lecture at MIT on Thursday. There were many things in the lecture that I’d not seen before – the one new idea I introduced here was the doubly infinite series . . . + 1 + 1 + 1 + 1 + . . . .

I wanted to use that series to illustrate some of the even stranger than strange properties these divergent infinite series have. This was a really fun part of our project today.

Saw this tweet from Fermat’s Library earlier today:

In 1895, Nikolay Bogdanov-Belsky painted the famous: "Mental Arithmetic. In the Public School of S. Rachinsky." The problem on the blackboard is (10²+11²+12²+13²+14²)/365, can you find a way to solve it with mental arithmetic? pic.twitter.com/PjhdQf4MAU

I was interested to see how the boys would approach this problem – and that was fun to hear. Unfortunately this project got way off the rails at the end when I tried to explain my solution to them. One of those unfortunate moments where something was clear to me but I was not communicating it well at all. I say it a lot in these blog posts – but they don’t all go well.

Here’s my older son’s solution:

Here’s my younger son’s version:

Next we move to me explaining my solution. This part did not go well at all:

Here’s the second part of the explanation of my solution. Neither kid likes my approach. Given the presentation, I’m not surprised at all

My younger son is working through a bit more of Art of Problem Solving’s Introduction to Geometry book this summer. Yesterday he came across a problem that have him a lot of trouble.

The problem asks you to prove that the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides.

Yesterday he worked through the solution in the book – today I wanted to talk through the problem with him. We started by introducing the problem and having my son talk through a few of the ideas that gave him trouble:

Next he talked through the first part of the solution that he learned from the book. We talked through a few steps of the algebra, but there were still a few things that weren’t clear to him.

Now we dove into some of the algebraic ideas that he was struggling with. One main point for him here, I think, was labeling the important unknowns in the problem.

For the last part, I wrote and he talked. I did this because I wanted him to be able to refer to some of our prior work. The nice thing here was that he was able to recognize the main algebraic connection that allowed him to finish the proof.

Last night I got an interesting comment on twitter in response to my Younger son suggesting that we write the numbers in a circle – a suggestion that we didn’t pursue:

At the beginning, there was a suggestion to write the numbers in a circle, then draw lines between the pairs. It is worth making a prediction about how that will look. The result may surprise you.

So, today we revisited the problem and wrote the numbers in a circle:

Next I asked them to try to find another set of numbers that would lead us to be able to pair all of the numbers together with the sum of each pair being a square. The discussion here was fascinating and they eventually found

This problem definitely made for a fun weekend. Thanks to Michael Pershan for sharing the problem originally and to Rod Bogart for encouraging us to look at the problem again using my younger son’s idea.

Yesterday I returned from a trip and the boys returned from camp, so we were together again for the first time in two weeks. I also happened to see this tweet from Michael Persian:

Shared this fantastic problem with counselors today.

This problem seemed like a nice one to use to get back in to our math project routine.

Here’s the introduction to the problem and the full approach the boys used to work through it the firs time:

When they solved the problem the first time around, they started by pairing 16 and 9. I asked them to write down their original pairs but to go through the problem a second time without starting with 16 and 9 and see if the choices really were forced. Here’s how that went:

This is a really nice problem for kids. It is easy to understand, so kids can jump right into it. There’s also lots of different ways to approach it. Definitely a fun way to get back into our math projects.

[had to write this in a hurry before the family headed off for a vacation – sorry that this post is likely a little sloppy]

Yesterday I gave a talk at a math camp for high school students at Williams College. The camp is run by Williams College math professor Allison Pacelli and has about 20 student.

The topic for my talk was the hypercube. In the 90 min talk, I hoped to share some amazing ideas I learned from Kelsey Houston-Edwards and Federico Ardila and then just see where things went.

A short list of background material for the talk (in roughly the order in the talk is):

(1) A discussion of how to count vertices, edges, faces, and etc in cubes of various dimensions

This is a project I did with my kids a few years ago, and I think helps break the ice a little bit for students who are (rightfully!) confused about what the 4th dimension might even mean:

(2) With that introduction I had the students build regular cubes out of the Zometool set I brought. Then I gave them some yellow struts and asked them to construct what they thought a hypercube might look like. From the prior discussion they knew how many points and lines to expect.

To my super happy surprise, the students built two different representations. I had my boys talk about the two different representations this morning. Funny enough, they had difference preferences for which was the “best” representation:

Here’s what my older son had to say:

Here’s what my younger son had to say:

At the end of this section of my talk I showed the students “Hypercube B” from Bathsheba Grossman (as well as my Zometool version):

(3) Now we moved on to looking at cubes in a different way -> standing on a corner rather than laying flat

I learned about this amazing way to view a cube from this amazing video from Kelsey Houston-Edwards. One of the many bits of incredible math in this video is the connection between Pascal’s triangle and cubes.

Here are the two projects I did with my kids a after seeing Houston-Edwards’s video:

After challenging the kids to think about what the “slices” of the 3- and 4-dimensional cubes standing on their corners would be, I showed them the 3D printed versions I prepared for the talk:

Part of my talk on hypercubes at Williams tomorrow -> 3D slices of a hypercube standing on a corner produce tetrahedrons and an octahedron pic.twitter.com/f7s8XM6kSB

I walked the students through how the vertices of a square correspond to the subsets of a 2-element set and then asked them to show how the vertices of a cube correspond to the subsets of a 3-element set.

There were a lot of oohs and ahhs as the students saw the elements of Pascal’s triangle emerge again.

Then I asked the students to find the correspondence between the 4-d cubes they’d made and subsets of a 4-elements set. I was incredibly happy to hear three different explanations from the students about how this correspondence worked – I actually wish these explanations were on video because I think Ardila would have absolutely loved to hear them.

(5) One last note

If you find all these properties of 4-D cubes as neat as I do, Jim Propp has a fantastic essay about 4 dimensional cubes:

By lucky coincidence, this essay was published as I was trying to think about how to structure my talk and was the final little push I needed to put all the ideas together.

The idea is easy to play with on your own -> deal out a standard deck of cards (arranged in any order you like) into 13 piles of 4 cards. By picking any card you like (but exactly one card) from each of the 4 piles, can you get a complete 13-card sequence Ace, 2, 3, . . . , Queen, King?

Here’s how I introduced Wright’s puzzle. I started the way he started – when you deal the 13 piles, is it likely that the top card in each pile will form the Ace through King sequence:

Now we moved on to the main problem – can you choose 1 card from each of the 13 piles to get the Ace through King sequence?

As always, it is fascinating to hear how kids think through advanced mathematical ideas. By the end of the discussion here both kids thought that you’d always be able to rearrange the cards to get the right sequence.

Now I had the boys try to find the sequence. Their approach was essentially the so-called “greedy algorithm”. And it worked just fine.

To wrap up, we shuffled the cards again and tried the puzzle a second time. This time it was significantly more difficult to find the Ace through King sequences, but they got there eventually.

They had a few ideas about why their procedure worked, but they both thought that it would be pretty hard to prove that it worked all the time.

I’m always happy to learn about advanced math ideas that are relatively easy to share with kids. Wright’s card puzzle is one that I hope many people see and play around with – it is an amazing idea for kids (and everyone!) to see.

My son stumbled on an amazing graph completely by accident the other day. He’s doing some work reviewing trig functions this week and I asked him to just play around with some graphs in Mathematica to get a feel for how Sin[x] and Cos[x] behave. One of the graphs he drew was:

from to :

I certainly wasn’t expecting him to make a graph like this one, but was happy that he did. Yesterday we talked through what was going on.

We started by discussing why the graph seemed so strange:

Now we dove into some of the details – which involve complex numbers and the definition:

as well as the definition of even and odd functions. So, there’s a lot of math to that we need to bring to the table to understand what’s going on in our graph.

Finally, we calculated the exact value of . Again, there’s a lot of advanced math that comes in to the calculation here – but even if some of the math ideas took a bit to sink in, I’d say that all in all it was a good conversation:

Last week I learned about an incredible fluid flow project from Jessica Rosenkrantz. I don’t want to give away the delightful surprise, so I’m not going to share the original tweets, but here is RosenKrantz’s pinned tweet – check her timeline for tweets on June 24, 2018 to see the inspiration for this project (and to see the idea we explore in today’s project executed to perfection):

For today’s project you need a little bit of paint (or s similar liquid), and two sheets of glass. Be careful doing this project with young kids – my kids are 12 and 14 and I thought they’d be able to handle the sharp glass themselves with gloves.

You spread out a bit of paint on one pane of glass, put the second pane on top, and then pull the two panes apart. What shapes do you expect to see in the paint after the panes are separated?

My younger son tried it out first:

My older son went second:

I think this is an amazing project to try with kids. Hearing what they think will happen and then hearing how they react to and describe what happens is really fun!