## A review of the game Pylos

Thanks to a tweet from Vincent Pantaloni last week I learned about the game Pylos:

This morning my younger son finished up his math work a little early so we decided to do a quick review of the game.

Here’s the game and my son’s thoughts about it after having played one game last night – I love that he sees that one of the math ideas involved in this game is sphere packing!

Next we demonstrated a game to show that the play itself is pretty easy for kids, yet filled with interesting strategy:

We wrapped up by discussing the game. I think my son’s comment that the game is surprisingly simple and fun is a great summary. Definitely a great game for kids.

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## Happy Pi Day

For Pi day today we explored the amazing near integer $e^{ \pi \sqrt{163}}$

I started by showing the boys the numbers as well as just how close it was to being an integer. I measured the closeness both in terms of the decimal expansion and in terms of the continued fraction expansion of the number:

Next I asked the boys to each take a turn finding another number relating to $\pi$ that was either nearly an integer or nearly a rational number. It turned out – especially with my younger son – to be a really nice way to discuss properties of powers of numbers.

The number my younger son found was $100 * 314^{\pi}$

The number my older son found was $3.4 * \pi^{\sqrt{2}}$

So – obviously just for fun – but still a neat way to talk about numbers and continued fractions. And a pretty fun number at the start, too 🙂

## A review of “The Sherlock”

We received a nice gift from Jim Propp earlier this month. With the kids off of school for a snow day today, it seemed like a good time to open it up.

My younger son played with it for a while and then I wanted to have him show how the game worked. Here are his initial thoughts about the game:

Here’s the first example of a puzzle solve. You’ll see that even having solved it once before it is still not necessarily so simple:

Here’s a second solve example – this one goes pretty quickly

Finally, we wrapped up by having him show some of the other pieces and me asking him to talk about what he thinks the main ideas are for this puzzle. Interestingly he doesn’t think that it is a math puzzle, but rather a logic puzzle 🙂

So, thanks to Jim Propp for giving us this really nice puzzle game.

## Sharing Stewart’s theorem with my son

My older son had a problem about finding the length of an angle bisector in a 3-4-5 triangle in his enrichment math class last week. Solving this problem is a little tedious, but also gives a great opportunity to introduce Stewart’s theorem. I first learned about Stewart’s theorem from Geometry Revisited when I was in high school. Here’s an explanation of the theorem on Wikipedia:

Stewart’s theorem on Wikipedia

I started off the project tonight by reviewing the original problem with my son:

Next I briefly introduced the theorem and then we got interrupted by someone knocking on our front door:

Now I showed how the proof goes. We had a brief discussion / reminder about the relationship between $\cos(\theta)$ and $\cos( 180 - \theta )$ and after that the proof went pretty quickly:

Finally, we returned to our original triangle to compute the length of the angle bisector using Stewart’s Theorem. The computation is still a little long, but now the calculations themselves are pretty straightforward:

Definitely a beautiful theorem. It is amazing that the law of cosines simplifies so nicely and that computing the lengths of cevians of a triangle.

## Sharing a neat problem from Matt Enlow with kids

Yesterday Matt Enlow shared a list of his 100 favorite problems:

I flipped through the problems yesterday and problem #6 struck me as a terrific one to share with kids:

I mentioned the problem to the boys yesterday and today we dove into it. Here are their initial thoughts:

Next I asked them to see if they could say anything at all about what would have to be true if there were powers of 2 and 3 that met the conditions of the problem.

My older son noticed a pattern in the powers of 2 mod 3. That helped us understand some basic ideas about what would have to be true if powers of 2 and 3 differed by 1. We then moved on from that idea to see how the “difference of squares” idea from algebra could help us show that the equation in the problem would probably never be true for an even power of 2 greater than 4. Nice start – now we just had to get to the finish line:

The idea that we were missing at the end of the last video was that powers of 3 only had 3 as a prime factor. Once the boys noticed that, they were able to see that an even power of 2 could never satisfy the equation!

Now we had to look at odd powers of 2. They noticed that roughly the same idea works if the power of 3 was even. There was one little subtle difference in the argument, but luckily both boys were able to explain that bit!

Now we had to look at the case with odd powers of 3 and odd powers of 2. Here I showed them how polynomials like $x^n - 1$ factor. I also shows how the numbers of the form $3^n - 1$ factor when n is odd.

The interesting idea here was that the factorization was always a 2 and an odd number. That showed the product could never be a power of 2. It took a while for us to get to that via the polynomial factoring, but we did get there.

Which then solved the whole problem!

Finally – just to wrap things up, I went to the computer to find powers of 2 and 3 that were “close” together using continued fractions:

I was lucky to see Matt Enlow’s list of problems on twitter yesterday. It is going to be a great resource for me – can’t wait to share more of the problems with the boys.

## Comparing a tetrahedron and a pyramid with theory and experiment

We’ve done a few projects on pyramids and tetrahedrons recently thanks to ideas from Alexander Bogomolny and Patrick Honner. Those projects are collected here:

Studying Tetrahedrons and Pyrmaids

One bit that remained open from the prior projects was sort of a visual curiosity. When you hold the zome Tetrahedron and zome Pyramid in your hand, it doesn’t look at all like the pyramid has twice the volume. Today’s project was an attempt to dive in a bit more into this puzzle.

We started by reviewing the ideas that Alexander Bogomolny and Patrick Honner shared:

Next we reviewed the geometric ideas that lead you to the fact that the volume of the square pyramid is double the volume of the tetrahedron.

Now we moved to the experiment phase – we put packing tape around the tetrahedron and the pyramid and filled them with water (as best we could). We then dumped that water into a bowl and used a scale to measure the amount of water. Our initial experiment led us to conclude that there was roughly 1.8 times as much water in the pyramid.

After that we repeated each of the measurements to get a total of 5 measurements of the volume of water in each of the shapes. Here are the results:

Definitely a fun project. I wish that we’d have gotten measurements that were closer to the correct volume relationship, but it is always nice to see that experiments don’t always match the theory!

## Sharring problem A1 from the 2017 Putnam with kids

We had a snow day today and I finally got around to sharing a neat problem from the 2017 Putnam Exam with the boys.

When I first saw the problem I thought it would be absolutely terrific to share with kids:

I started off the project today by having them read the problem and spending a little bit of time playing around:

After the initial conversation the boys, I triehd to start getting a bit more precise. The first sequence of numbers they knew was in the set was 2, 7, 12, 17, . . . .

They were not sure if 4 was in the set or not. My first challenge problem to them was to show that if 4 was in the set, then 3 would be in the set.

My next challenge question was whether or not 1 would be in the set.

Now we moved on to one of the number theory aspects of the problem – is 5 in the set?

During this conversation my younger son noticed that we had found a number that was 1 mod 5.

Finally, we talked through how you could find 6 from the number my son noticed in the last video.

I’m really happy with how this project went. This problem is not one (obviously) that I would expect the kids to be able to solve on their own, but most of the steps necessary to solve the problem are accessible to kids. It was really neat to hear their ideas along the way.

## Sharing “developable surfaces” with kids thanks to a brilliant lecture from Heather Macbeth

[This is a redo of a blog post from January 2018 that somehow ended up 1/2 deleted. Not sure what I did to that old post, but I didn’t want to lose the ideas.]

In January 2018 I attended a terrific public lecture given by Heather Macbeth at MIT. The general topic was differential geometry, and the specific topic she discussed was “developable surfaces.”

Here’s an example from the talk:

I also printed a few examples and shared them with the boys the next day:

Here’s what my older son had to say about the shapes:

Here’s what my younger son thought:

These are really neat surfaces to explore. If you look at some of the mathematical ideas for “Developable surfaces” you’ll find that some of the surfaces are actually pretty easy to code, print, and share with kids!

## Sharing a great Alexander Bogomolny probability problem with kids

[note: I’m trying up this post at my son’s karate class. It is loud and unfortunately I forgot my headphones. I’m left having to describe the videos without being able to listen to them . . . . ]

I saw a really great problem today from Alexander Bogomolny:

By coincidence I heard the recent Ben Ben Blue podcast yesterday which had a brief mention / lament that it was hard to share mistakes in videos.

This problem is probably a good challenge problem for my older son and definitely above the level of my younger son. But listening to both of them try to work through the problem was really interesting.

I started with my older son – he initially approached the problem by comparing the individual probabilities:

After his initial work, I talked with him about comparing the probabilities of the complete events described in the problem. Initially there was a little confusion on his part, but eventually he understood the idea:

Next up was my younger son – not surprisingly, he had a hard time getting started with the problem. His initial approach was similar to what my older son had done – he looked at the one head and two heads events separately to see which one was more likely for each coin:

As I did with my older son, I asked him to look at the two events as a single event and see which one was more likely when each coin went first:

So, a nice project and an opportunity to see a few mistakes and as well as how kids approach a challenging probability problem.

## Comparing double stuffed Oreos to thin Oreos

In 2015 we did a project comparing double stuffed oreos to regular ones:

Do double stuffed oreos have double the stuffing?

As I said in that blog post, I’d seen a few teachers discussing the idea, but I don’t remember who originally shared the project. So, to be clear again, the idea for this line of study isn’t mine, but I’m happy to have some fun with it.

Instead of revisiting the prior project today, I tried something slightly differnt -> comparing double stuffed oreos with thin ones. The prep work for this project proved to be a little harder than I was expecting because the double stuffed oreo shells were really fragile. So, if you want to repeat this project, be prepared for lots of broken oreo shells!

To start I introduced my son to the problem we were going to try to solve today and asked for his thoughts. The problem was to try to find the ratio of the volume of stuffing in the double stuffed oreos to the volume of stuffing in the thin ones.

Our original intention was to weigh 10 of the crackers from each of the 2 types of cookies. We were able to get only 8, though. The 8 thin crackers weighted 22 grams and the 8 double stuffed crackers weighed 33 grams.

Sorry the writing was off screen.

Next we moved to weighing the full cookies. I didn’t communicate really well at the start of this video, and confused my son a bit. Eventually we got back on the same page weighed 4 cookies of each type.

The 4 double stuffed cookies together weighed 60 grams. That led us to conclude that the stuffing weight was approximately 27 grams for 4 cookies.

Finally, we repeated the process in step 3 with 4 thin oreos. We found that the 4 cookies together weighed roughly 30 grams, meaning the total weight of the filling was 8 grams.

So, my son’s guess of 4 to 1 for the ratio of the filling weight was pretty close. Turned out be 27 grams to 8 grams for 4 cookies, or about 3.5 to 1.

Definitely a fun project. I haven’t done much in the way of introductory statistics for kids – this project definitely gets kids engaged!