# Labeling each vertex of a permutahedron is a terrific mathematical exercise for kids

Yesterday we did a fun project exploring a permutohedron:

Last night I thought it might be neat to have the kids try to label the vertices of a permutohedron with the permutations represented by each vertex. Fortunately, it was possible to build a truncated octahedron with the green Zometool struts:

We started out today’s project by talking about the rules for making a permutohedron in different dimensions. Here I used the labeling of the permutation of 3 objects as a base case to make sure the boys understood the directions properly.

Next I had the boys label each vertex of the permutahedron with the permutation of {1,2,3,4} that the vertex represented. Then, they talked about the process of figuring out the right labels.

I’m sorry that the video below runs 10 min, but if you listen to the whole discussion I think you’ll see that seemingly straightforward act of labeling these vertexes is a terrific mathematical exercise for kids.

# A morning with the permutohedron

Today we are revisiting an old project on a really neat shape -> the permutohedron:

“A fun shape for kids to explore – the permutohedron

I learned about this shape thanks to Allen Knutson at Cornell – he included a fun pic of a large permutohedron in the comment of the blog post above:

He also pointed me to a 3d print on Thingiverse that we used in the last project and again today:

“Permutahedron” by PFF000 on Thingiverse

So, I started today by having the boys describe the 3d printed shape. We have two versions – a larger one that unfortunately broke a little and a smaller – but in one piece! – version. Here’s what the boys had to say about the shapes:

Next I had the boys read the Wikipedia page on the permutohedron for about 10 min and then we discussed some of the ideas that they thought were interesting:

Finally, we built the 2-D permutohedron and showed how it was embedded in a 3d grid:

Definitely a fun project and it is always great to be able to have kids hold interesting math ideas in their hands!

# Exploring introductory trig using 3d printing

My younger son is studying trig right now and I thought it would be fun for him to play around with some 3d curves made with trig functions.

I showed him how to use Mathematica’s ParametricPlot3D[] function and then just let him make some shapes on his own. He settled on a curve that looked like this:

Here’s the code just in case it is not legible in the video:

After he made the curve we printed it – it was really fun to see him working on the print when it was finished. I wish the picture was better!

When everything was finished I asked him to tell me about the curve. I’d not seen the code before and didn’t know there was a stray Cos[x] in it. Talking about that piece of the code led to a great conversation about elementary trig functions (totally by accident!):

I really enjoyed this project today – it is fun to use 3d printing to explore so many different areas of math.

# Sharing Ricky Reusser’s ‘Periodic Planar Three-Body Orbits” program with my son

I saw an really neat idea in a tweet from Nalini Joshi yesterday:

A direct link to Ricky Reusser’s incredible 3-body problem visualization is here:

Ricky Reusser’s amazing 3-body problem visualization

For today’s math project I asked my son to play around with the program and pick three examples that he found interesting. The discussion of those three examples is below.

Here’s the first one, with a short discussion of three body problem at the start:

Next up was an orbit shaped almost like an infinity symbol:

Finally, an orbit that it completely amazing – I almost can’t believe a shape like this is possible!

# Sharing Problem 10 from Mosteller’s 50 Challenging Problems in Probability with my younger son

Today my older son is at an event, so the project was just with my younger son. The project is a version of a famous problem attributed to Daniel Ellsberg (Mosteller’s book also credits Ellsberg). Here’s the Wikipedia page on the more famous problem:

I got start with today’s project by reading the first problem and having my son share his initial thoughts about how to approach solving it:

With some thoughts down on the white board, now we turned to solving the problem. I loved his thought process here:

Next we moved on to problem attributed to Ellsberg – this gives problem is a fun twist on the first problem since now you do not know how many balls of each color are in the bag:

With my son having written down a few ideas about the new problem in the last video, he now gave his solution to the 2nd problem. His thinking here is also really great. It is fun to see young kids talk through a difficult problem like this one:

# What are the chances of a class with 24 students having 3 pairs of students sharing a birthday?

Yesterday we looked at the famous Birthday problem – how many people do you need to have in a room to have a 50/50 chance of two people having the same birthday? That project is here:

Diving into the Birthday problem with kids

Today we continued the project (with just my older son as my younger son was hiking) and studied the problem that originally motivated this project -> If you have 24 students in a class, what is the chance that exactly 3 pairs of students will share a birthday? This is the surprisingly fun situation in my son’s English class.

We will – as I think it standard for the introductory version of this problem – be making the assumption that all birthdays are equally likely. If you want to see a really neat discussion – though not really a math for kids paper! – see the paper in this tweet:

So, to start the project today we first reviewed the main ideas from yesterday:

Next we took a step towards solving the problem by looking at the chance of having exactly 2 pairs. Once piece of the counting here is tricky, so we used the computer to help see what the problem was.

Now we tackled the “exactly 3 pairs problem”:

Finally, I had my son make up a problem to solve – he decided to find the chance of all 24 students pairing up. This problem wasn’t too hard given the prior work. It was also a fun challenge to try to estimate the chance of this happening.

# Diving into the Birthday problem with kids

Earlier in the week I learned that my older son’s high school English class has 24 students and 3 pairs of students who share the same birthday. None are twins, so no tricks or anything like that, just a fun fact for this particular class.

I thought it would be fun to figure out how rare something like this would be – assuming, of course, that all of the birthdays are randomly distributed amount the 366 possible birthdays (366 because many of the kids were born in 2004).

It turns out the chance of having exactly three pairs of kids with the same birthday (and no other shared birthdays) in a class of 24 kids is roughly 2.3%, or if you prefer the exact answer:

Instead of continuing with Mosteller’s book this weekend, I thought it would be fun to dive into the birthday problem. I started today with the standard problem – how many people do you need in a room for a 50% chance of two people sharing the same birthday. This is not an easy problem and the answer is not intuitive.

Here’s how we got started – not surprisingly, down a path that wasn’t quite right:

After coming up with a formula in the last video, we went to Mathematica to see what it said. Here we discovered that the formula was giving answers that were not correct:

Now we returned to the whiteboard and the boys found a new formula – this one calculated the chance of having exactly 1 pair with the same birthday. I was happy that they were able to derive this formula and even happier for the chance to show them it didn’t agree with our computer modelling!

Now we went back to the computer to see the surprise that our new – and much closer to correct – formula actually didn’t agree with the modelling. What was wrong?

Finally, having figured out why the two approaches didn’t match, the boys were able to find the correct formula to solve the problem. Tomorrow we’ll dive into the more complicated problem of finding the probability of 3 pairs:

# Sharing problem #9 from Mosteller’s 50 Challenging problems in Probability with kids -> how to play craps!`

Problem #9 from Mosteller’s 50 Challenging Problems in Probability is about the game of craps. The question asks, essentially, does the player of the casino have a better chance of winning the game.

This is both a fun and reasonably difficult problem for kids, but it led to a terrific conversation.

Here’s how I introduced the problem:

Following the introduction, I had the boys solve for the probabilities of an immediate win or loss:

Now we moved on to the harder question – what happens if you roll. say, and 8 on the first roll. How do we find the probability that you win the game in this situation?

One way would be summing an infinite series, but I hoped to introduce the boys to a simpler way of seeing the probability here:

Having solved one of the hard cases exactly in the last video, we moved on to solve the rest of them here:

Finally, we went to Mathematica – not for anything super complicated, just to add up the fractions – so we could find out whether or not the player or the casino had the advantage in the game:

This problem is a great one for kids to explore – it really shows how a systematic approach to problem solving can help you get throw a pretty challenging problem.

# Sharing problem #8 from Mosteller’s 50 Challenging Problems in Probability with kids

We are up to problem #8 in Mosteller’s 50 Challenging Problems in Probability.

The problem today is a classic -> What is the probability that you would be dealt a bridge hand with 13 cards of the same suit from a well-shuffled deck of cards?

We started off by taking a quick look at the problem and getting a few ideas from the boys about how to solve it. One question that came up was whether or not the method used to deal the cards would matter:

The first cut at solving the problem involved dealing the cards in a circle – so what I’d think of as the standard way to deal cards:

Next up we took a little detour into choosing numbers because some of the details of how those numbers worked were a little fuzzy. It was a nice review and I was happy that the boys had recognized that the choosing numbers were somehow related to problem:

Finally, we wrapped up by checking to see if the probability changed when we used a different method of dealing.

I think this is a great question for kids to think through. The thoughts from the boys here are probably representative of some of both the struggles and connections that kids will have thinking through this problem.

# Sharing Vladimir Bulatov’s Tetrahedral Limit Set with kids

Last week I saw a really neat tweet shared by Alex Kontorovich:

I ended up buying Bulatov’s piece from Shapeways and it came today. Here’s a quick video look at it:

When the kids got home from school I asked them to take a look at it and share their thoughts.