A neat unsolved problem in number theory that kids can explore

Yesterday I saw a really neat thread on the Collatz conjecture from Alex Kontorovich

In that thread is a blog post by Alex’s friend Igor Park and Park’s blog post as a link to a neat set of lecture notes by Barry Mazur. AND, in Mazur’s notes is this “new to me” unsolved problem in number theory:

Mazur

Instead of continuing on our journey through Mosteller’s 50 Challenging Problems in Probability, I decided to explore this problem with the boys today.

Here’s the introduction to the problem and a bit of playing around with a few of the small cases:

In the last video the boys thought that the squares would all have to be odd and the primes would have to be odd. Here we explored both of those conjectures. That exploration led to a discussion of why odd numbers always have squares that are congruent to 1 mod 8:

Now we continued the discussion from last video and investigated the primes that could appear in this problem. We started by showing that 2 could never appear and then eventually found that only primes of the form 4k + 1 could appear:

Next we moved to the computer to explore more cases of the conjecture. This was mainly an exercise into writing a simple program in Mathematica, but it led to an interesting discussion as well as an idea for further exploration:

Finally, we modified our program to explore the number of different solutions to the problem for each number. The modification to the program was actually really easy and the histogram was fascinating to see:

It is really fun to be able to explore an unsolved problem with kids. I especially love unsolved problems that allow kids to get in some secret arithmetic practice will getting a bit of exposure to some advanced ideas in math. Seeing this problem yesterday and getting to explore it today with the boys was a real treat!

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Stumbling through problem #4 in Mosteller’s 50 Challenging Problems in Probability

Sometimes I think a project is going to go really smoothly and I’m just plain wrong. Today was one of those days, unfortunately, as I completely misjudged how difficult this problem would be for my younger son.

He and I ended up spending another 20 min on the problem after the project was over and that time was much more productive. I’m kicking myself a little – and wish that I’d approached the problem differently – but you can’t win them all 🙂

With that disclaimer out of the way, problem #4 from Mosteller’s probability book is a classic:

How many rolls, on average, does it take to roll a 6 on a fair, 6-sided die?

Here’s the introduction to the problem and the initial thoughts from both kids:

Next I had the boys roll dice off camera and record how long it tool to roll a 6. Here are the results of those experiments:

Now we moved from experiments to diving into the math – this is where I probably should have realized that my younger son was struggling a bit to see the math, but I failed to see his struggle:

And then we get lost, unfortunately. I turn the camera off around 6 min and we spent 10 more min talk about the problem off camera. I share this video only so that I can go back and learn from it later and see what I could have done better. Not everything goes well all the time . . . .

While we spoke off camera my older son found a very clever way to solve the problem. Here he explains that solution and my younger son was able to chime in on one little mistake at the end. It was a nice silver lining to a project that went off the rails a little bit:

Sharing problem #3 from Mosteller’s 50 Challenging Problems in Probability with kids

We are working through Mosteller’s 50 Challenging Problems in Probability this fall. Today we tackled problem #3 which is an problem that is definitely accessible to kids and has a fun and surprising result.

The problem can be summarized like this:

You have two people who each have probability p of making a correct decision and a third person who just flips a coin. If this group of three reaches a decision by majority rule, what is the probability that they make the correct decision?

I started the project today by sharing this problem with the boys and asking them what they thought:

They had a pretty good idea about how to approach the problem, so for the 2nd part of the project we dove in to the calculation and found the surprising result:

To wrap up, we wrote a short computer program to simulate the problem and see if the results of that program matched what we’d found in the second part:

I really like this problem – easy for kids to understand, not too difficult to compute the answer, the computation allows a few different approaches and lessons about probability, and the result has a nice surprise! Fun project 🙂

Having the boys talk through Problem #2 from Mosteller’s 50 Challenging Problems in Probability

This fall we are going through Frederick Mosteller’s 50 Challenging Problems in Probability. Our first project was last week, today we tackled problem #2. The problem goes like this:

You are challenged to win two games in a row in a series of three games against opponent A and B. B is a better player than A. You can choose to play the the three games in order ABA or BAB – which order gives you a better chance of winning two games in a row?

Here’s the problem and some initial thoughts from the boys – they found the problem to be pretty challenging:

Next we chose some specific probabilities of winning the games against player A and against player B and calculated the exact probability of winning two games in a row in each case:

Having worked through a specific calculation, next we solved the problem in general:

Finally, we went to Mathematica to run some simulations and see if our results matched with theory (and also to introduce the boys to some basic logical operations in programming):

A linear algebra follow up to our probability project

My older son is learning linear algebra right in Strang’s Linear Algebra book. Over the weekend we did a fun project on problem number #1 in Frederick Mosteller’s probability challenge book and I wanted to show him a neat follow up involving linear algebra today.

The original project is here:

Going through problem #1 from Frederick Mosteller’s probability challenge book with kids

Today what I wanted to show my son is how you solve the recurrence relation for the terms we found in the original project. It is a fun linear algebra example.

We started by reminding ourselves what the terms were and then looking at the recurrence relation we found on the Internet Sequence Database:

By the end of the last video we had a cubic equation written down and in this video my son worked to find the roots of that cubic:

Now we looked at linear combinations of the powers of the roots that we found in the last video to see if we could find the general solution. The general solution involved a matrix equation:

We did not solve the matrix equation by hand, but rather went to Mathematica to save time.

Once we solved the equation we saw that we could find any term in the sequence that we wanted!

Following up yesterday’s probability problem with one of my favorite ideas involving high school algebra

Yesterday we did a neat project on problem #1 from Frederick Mosteller’s probability challenge book:

Going through problem #1 from Frederick Mosteller’s probability challenge book with kids

Today my older son is off mountain biking so the follow up project is with my younger son who is in 8th grade. I thought it would be fun to look at additional solutions to yesterday’s puzzle and show him how we could write down a formula for those solutions.

We started by look at some of the small solutions to yesterday’s problem and looking for patters. My son noticed a connection to \sqrt{2} that made me really happy!

Next I showed him the Internet Sequence Database. I wanted to show him that sometimes when you are looking at a sequence of integers, it is something that other people have studied before:

Now we returned to the whiteboard to study the sequence more carefully. Our starting point was the recurrence relation that we learned about in the last video:

Finally, and this is one of my favorite high school algebra examples, we took a first step at solving the recurrence relation. This step is a nice application of factoring and using the quadratic formula:

Going through problem #1 from Frederick Mosteller’s probability challenge book with the boys

About a week ago I saw this neat tweet:

So, I bought the book (plus one other from the thread!):

My plan for this fall is to go through as many of the 50 challenging probability problems as we can. I don’t know how many are accessible to the boys, but hopefully most of them are with some help.

Today we tackled problem #1 – You have some red socks and some black socks in a drawer. When you pick two socks at random the probability of a red pair is 1/2. What is the smallest number of socks that could be in the drawer?

Here’s how I introduced the problem to the boys and their initial solution:

The second part of the problem asks what the minimum number of socks is assuming that the number of black socks is even. This problem gave the boys a bit more trouble, but was a great learning opportunity for them.

In the last video they showed that the number of black socks couldn’t be 2 or 4 with direct computation. Here they showed that it could be 6 using algebra. This was a nice opportunity for some factoring practice.

Finally, we went to the computer and wrote a short (and obviously inefficient) program to test out some solutions. It was fun to find a few more (sadly we were super rushed for time here, but still had a nice conversation).