Jim Propp’s “Swine in a line” game part 2

Last week I saw a really fun new question from Jim Propp:

Here’s the first project that we did on the game:

Jim Propp’s Swine in a Line game

Today we returned to the game to see if we could make any more progress understanding how it worked.

First we reviewed the rules and decided on an initial approach to studying the game for today:

Their first idea was to try to keep the two ends open since they knew the result when you reached the position with only 1 and 9 open.

Now we tried to study a bit more. The kids were having trouble seeing a path forward, so I just let them play.

At the end of the last video we were studying a position with all of the pens filled except for 7 and 9. In this video we searched for a winning move.

Finally, we took one more crack at solving the game. They boys got very close to the main idea, about an inch away(!), but didn’t quite get over the line.

The boys were really interested in the game and we kept talking for about 30 min after the end of the project. During that talk they did uncover the main idea. After that we played several games where they followed the strategy and they were able win against me every time following that strategy. It was a fun way to end the morning.

Jim Propp’s “swine in a line game”

Saw this great new video from Jim Propp yesterday:

This morning I had the boys watch the video and then we spent maybe 15 min talking through the game and seeing what we could learn.

First I asked them what they thought after seeing the video:

Now we played the game and the boys made a couple of initial discoveries. You can see quickly why this is a fun game for kids to play around with:

Next we played the game one more time. We aren’t trying to solve the game in this project, just to try to learn a few things about it.

Finally, we wrapped up by talking about a few of the things they learned playing the game. This part didn’t quite go how I wanted, but it was still interesting to hear what they had to say.

I’m excited to play around with this game a bit more later in the week. It’ll be interesting to see if the boys can continue to make project towards the solution.

Playing with Dan Anderson’s complex map program

Dan Anderson made a really neat little complex mapping program today:

This program allowed us to do a fun continuation of the project that we did over the weekend:

Sharing Kelsey Houston-Edwards’s Complex Number video with kids

Looking at the complex map z -> z^2 with kids

So, with each kid tonight I had them just play with the program and then I played a game of tic tac toe with them for fun 🙂

Here’s my older son playing:

and here’s our tic tac toe game:

Here’s my younger son playing:

Finally, here’s our tic tac toe game

Definitely a fun project. Can’t wait to play with the program more – this is a really fun subject to share with kids.

Looking at the complex map z -> z^2 with kids

Yesterday we did a fun project using Kelsey Houston-Edwards’s compex number video:

Sharing Kelsey Houston-Edwards’s Complex Number video with kids

The boys wanted to do a bit more work with complex numbers today, so I thought it would be fun to explore the map $Z \rightarrow Z^2.$ The computations for this mapping aren’t too difficult, so the kids can begin to see what’s going on with complex maps.

We started by looking at some of the simple properties. The kids had some good questions right from the start.

By the end of this video we’ve understood a bit about what happens to the real line.

After looking at the real line in the last video, we moved on to the imaginary axis in this video. The arithmetic was a little tricky for my younger son, so we worked slowly. By the end of this video we had a pretty good understand of what happens to the imaginary axis under the map $Z \rightarrow Z^2.$

At the end of this video my younger son noted that we hadn’t found anything that goes to the imaginary axis. My older son had a neat idea after that!

Next we looked at $(1 + i)^2$. We found that it did go to the imaginary axis and then we found two nice generalizations that should a bunch of numbers that map to the imaginary axis.

Finally, we went to Mathematica to look at what happens to other lines. I fear that my attempts to make this part look better on camera may have actually made it look worse! But, at least the graphs show up reasonably well.

It was fun to hear what the boys thought they’d see here versus their surprise at what the actually saw 🙂

I think this is a pretty fun project for kids. There are lots of different directions we could go. They also get some good algebra / arithmetic practice working through the ideas.

Sharing Kelsey Houston-Edward’s complex number video with kids

I didn’t have anything planned for our math project today, but both kids asked if there was a new video from Kelsey Houston-Edwards! Why didn’t I think of that 🙂

The latest video is about the pantograph and complex numbers:

Here’s what the boys thought about the video:

They boys were interested in the pantograph and also complex numbers. We started off by talking about how the pantograph works. With a bit more time to prepare (and probably a bit more engineering skill than I have), building a simple pantograph would make a really fun introductory geometry project.

Next we talked about complex numbers. We’ve talked about complex number several times before, so the idea wasn’t a new one for the boys. I started from the beginning, though, and tried to echo some of the introductory ideas that Kelsey Houston-Edwards brought up in her video.

To finish up today’s project we looked at some basic geometry of complex numbers. The specific property that we looked at today was multiplying by i. At the end of this short talk I think that the boys had a pretty good understanding of the idea that multiplying by i was the same as rotating by 90 degrees.

Complex numbers are a topic that I think kids will find absolutely fascinating. I don’t know where (if at all) they come into a traditional middle school / high school curriculum, but once you understand the distributive property you can certainly begin to look at complex numbers. It is such a fun topic with many interesting applications and important ideas – many of which are accessible to kids. Just playing around with complex numbers seems like a great way to expose kids to some amazing math.

A night with Cut the Knot, Nassim Taleb, and some Supernova

Please note the correction at the bottom of the post

A further correction – there is still an error. Ugh. This approach may not work, unfortunately . . .

Saw a neat problem from Alexander Bogomolny earlier today:

I actually missed the problem when it was initial posted, but saw it via Nassim Taleb’s solution:

The problem sort of gnawed at me all day and I figured it was in the maybe 1 in 10 problems that Bogomolny posts that I might be able to solve.

So, tonight I poured a glass of Supernova and gave it a go

One thing on my mind all day with this problem was Jensen’s inequality. What I would *love* to be able to do is say that by Jensen’s inequality:

$(1/3) \sqrt{x^2 + 3} + (1/3) \sqrt{y^2 + 3} + (1/3) \sqrt{xy + 3}$

$\geq \sqrt{ (1/3)( x^2 + y^2 + xy) + 3}$

Which is easily seen to be $\geq 2$ because of the constraint $x + y = 2.$ That work would show the original inequality was $\geq$ 6.

The approach has a tiny bit of merit since $\sqrt{x^2 + 3}$ is concave up for $x$ between 0 and 2 -> here’s a little Mathematica plot showing that the second derivative is indeed positive on 0 to 2:

But . . . the problem is that folding in the 3rd term in the sum is stretching the rules of Jensen’s inequality a bit, I think, since it is not of the form $\sqrt{a^2 + 3}$.

With the first two terms, though, applying Jensen’s inequality seems ok, but I now need (1/2)’s instead of (1/3)’s since there are only two terms. So, I’ll use Jensen’s on the first two terms only and try to show that

$(1/2) \sqrt{x^2 + 3} + (1/2) \sqrt{y^2 + 3} + (1/2) \sqrt{xy + 3} \geq 3$

By Jensen’s inequality this new sum is

$\geq \sqrt{ \frac{x^2 + y^2}{2} + 3} + (1/2) \sqrt{xy + 3}$

A bit of algebra and the fact that $x + y = 2$ allows us to simplify this expression to:

$\sqrt{5 - xy} + (1/2) \sqrt{xy + 3}$

and then further to:

$\sqrt{ (x - 1)^2 + 4} + (1/2) \sqrt{4 - (x - 1)^2}$

Now we are just down to a fairly straightforward calculus problem, and I’ll let Mathematica do the heavy lifting since the algebra isn’t that interesting:

We can see visually that the minimum occurs at $x = 1$ from the plot, and the plot of the derivative further confirms that there is only one critical point. The value of the last expression at $x = 1$ is indeed 3 as we were hoping.

So, Jensen’s inequality, a bit of calculus, and a nice glass of scotch shows that the original inequality is indeed true.

Thanks to Alexander Bogomolny for the problem, and to Nassim Taleb for his solution that got me thinking about the problem.

Correction

I received a note from Alexander Bogomoly over night. He spotted an error in the calculation:

and I thought my kids having trouble sleeping and waking me up at 5:00 am today was a bad start to the day!

But it seems that I’ve gotten very lucky – both learning from my carelessness in applying Jensens inequality and that the path forward from Bogomolny’s correction is easier than the path I actually took.

Starting here – we wish to show that:

$(1/2) \sqrt{x^2 + 3} + (1/2) \sqrt{y^2 + 3} + (1/2) \sqrt{xy + 3} \geq 3$

The correction shows that the expression on the left hand side is $\geq$ than

$\sqrt{ (\frac{x + y}{2})^2 + 3} + (1/2) \sqrt{xy + 3}$

but since $x + y = 2$, the first piece of this expression is equal to 2 and the 2nd expression simplifies as before. So we are left with

$2 + (1/2) \sqrt{4 - (x - 1)^2}$

and this expression has a maximum of 3 at $x = 1.$

That means that the expression we were trying to show to be greater than 3 is indeed greater than 3, and the expression in the original tweet is greater than 6.

I’m grateful to Alexander Bogomolny for spotting the error in my original argument.

Playing with 4d Toys

Quick post tonight because I’m running out to dinner . . . .

Here’s a link to their site:

The 4D Toys site

It is a nice compliment to some of the 4th dimensional projects we’ve been doing. Here’s what my younger son thought after playing around with it for a bit:

Here’s what my older son thought after playing with it for 10 min:

Excited to use this app a bit more!

A good (though tricky) introductory counting problem

My older son is re-working his way through Art of Problem Solving’s Introduction to Counting and Probability. He came across a problem in the review section for chapter 5 that gave him some trouble. I decided to talk through part of it tonight and included my younger son.

My younger son hasn’t been studying any counting lately, so I was expecting the problem to be pretty challenging for him. His work through the first part of the problem is, I think, a nice example of a kid working through a challenging math problem.

The problem is this: How many different ways are there to put 4 distinguishable balls into 3 distinguishable boxes?

The next problem is what was giving my son some trouble: How many different ways are there to put 4 distinguishable balls into 3 indistinguishable boxes?

My younger son struggles with the problem for a bit on this one and then my older son offers his thoughts. What gives my older son a little trouble is the case in which you put 2 balls into one box and 2 into another.

So, after struggling with 2-2-0 case in the last video, we talk about it in a little more depth here. The tricky part is seeing that two cases that don’t look the same are actually the same. It was harder for the boys to see the over counting than I thought it would be. But, we made it!

So, the indistinguishable counting part was pretty confusing to the boys. I think we need to do a few more problems like this to let this particular counting concept sink in.

One more look at the hypercube

We’ve done two projects on hypercubes after seeing Kelsey Houston-Edwards’s latest video. Those two projects are here:

Kelsey Houston-Edwards’s hypercube video is incredible!

Revisiting Kelsey Houston-Edwards’s hypercube video

The video that inspired those two projects is here:

After posting the second project our friend Roy Wiggins shared a video that he made several years ago:

After seeing Wiggins’s video I thought it would be fun to use in one more project with the boys. While they were at school today I printed the set of shapes corresponding to the intersection of the hypercube and 3d hyperplanes that you can see in both videos above. When the boys got home I talked about the shapes with them individually. In addition to the 3d prints, I had some zometool shapes set up to help understand the more traditional way to view and understand a hypercube.

Here’s how the talk went with my younger son:

And here’s what older son and I talked about:

This was definitely a fun series of projects. It was interesting to me that both kids really struggled to see how to explain the shape of a hypercube from these 3d shapes. Not that I was expecting it to be easy (!) but this alternate view of the hypercube proved to be much more difficult to process than I expected.

Revisiting Kelsey Houston-Edwards’s Hypercube video

Last week Kelsey Houston-Edwards published a fantastic video about hypercubes – it is one of the best math videos I’ve ever seen:

Here’s our project on her video:

Kelsey Houston-Edwards’s hypercube video is incredible!

Today while the kids were at school I wrote a couple of Mathematica functions to replicate her results. Writing the code to make these shapes is actually a pretty fun exercise in linear algebra and trig, but that’s a little more than I felt like sharing with the kids just now 🙂

Instead I had them look at the shapes on the screen and tell me what they thought. The first video in each with each kid shows the shape made by a plane intersecting a 3d cube standing on its corners. The second video shows the 3d intersection of a hyperplane perpendicular to the long diagonal of a 4d cube intersecting that cube.

Here’s what my older son had to say:

(a) The 3d cube being cut by a (slightly thick) plane:

(b) The 4d cube being cut by the hyperplane:

Here’s what my younger son had to say:

(a) The 3d cube being cut by a (slightly thick) plane:

(b) The 4d cube being cut by the hyperplane:

This project was really fun, and, as I mentioned above, would also be a great programming project for kids learning linear algebra and trig. I’m 3d printing some of the shapes how, so playing with those shapes will be our project tomorrow!