Tag number theory

Introducing the basic ideas behind quadratic reciprocity to kids

We are heading out for a little vacation before school starts and I wanted a gentle topic for today’s project. When I woke up this morning the idea of introducing the boys to quadratic reciprocity jumped into my head. The Wikipedia page on the topic gave me a few ideas:

Wikipedia’s page on quadratic reciprocity

I started the project by showing them the chart on Wikipedia’s page showing the factorization of n^2 - 5 for integers 1, 2, 3, and etc . . .

What patterns, if any, would they see?

Next we moved to a second table from Wikipedia’s page – this table shows the squares mod p for primes going going from 3 to 47.

Again, what patterns, if any, do they notice?

Now I had them look for a special number -> for which primes p could we find a square congruent to -1 mod p?

Finally, we wrote short program in Mathematica to test the conjecture that we had in the last video.  The conjecture was that primes congruent to 3 mod 4 would have no squares congruent to -1 mod p, and for primes congruent to 1 mod 4 would, -1 would  always be a square.

Sorry for the less than stellar camera work in this video . . . .

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Trying out Edmund Harriss’s puzzle with kids

Saw a neat puzzle posted by Edmund Harriss last night:

I thought it would be fun to try it out with the boys this afternoon.

I didn’t give them much direction after introducing the puzzle – just enough to make sure that my younger son understood the situation:

After the first 5 minutes they had the main idea needed to solve the puzzle. In this video they got to the solution and were able to explain why their solution worked:

Definitely a fun challenge problem to share with kids. You really just have to be sure that they understand the set up and they can go all the way from there.

Walking through the proof that e is irrational with a kid

My son is finishing up a chapter on exponentials and logs in the book he was working through this summer. The book had a big focus on e in this chapter, so I thought it would be fun to show him the proof that e is irrational.

I started by introducing the problem and then with a proof by contradiction example that he already knows -> the square root of 2 is irrational:

Now we started down the path of proving that e is irrational.  We again assumed that it was rational and then looked to find a contradiction.

The general idea in the proof is to find an expression that is an integer if e is irrational, but can’t be an integer due to the definition of e.

In this part we find the expression that is forced to be an integer if e is irrational.

Now we looked at the same expression that we studied in the previous video and showed that it cannot be an integer.

I think my favorite part of this video is my son not remembering the formula for the sum of an infinite geometric series, but then saying that he thinks he can derive it.

This is a really challenging proof for a kid, I think, but I’m glad that my son was able to struggle through it. After we finished I showed him that some rational expressions approximating e did indeed satisfy the inequality that we derived in the proof.

Revisiting the last digits of Graham’s number

Several years ago we did a bunch of projects on Graham’s number.

An attempt to explain Graham’s number to kids, and

The last 4 digits of Graham’s number

These projects were inspired by this fantastic Evelyn Lamb article:

Graham’s numbers is too big for me to tell you how bit it is

Today I thought it would be fun to revisit the calculation of a few of the last digits of Graham’s number.

So, with no review, I asked my older son what he remembered about Graham’s number and then we talked about the surprising fact that you could calculate the last few digits even though you really couldn’t say much else about the number:

Next I asked my son about how he would approach calculating the last digit. He gravitated to the right idea -> modular arithmetic. The ideas were a little confusing to him, but I let him work mostly on his own.

We didn’t get to the end in this video, but you can see how the ideas start coming together.

In the last video he had made some progress on finding the last digit, but one piece of the argument kept giving him trouble.

BUT, he did have a correct argument – it just took him a minute to realize that he was on the right track.

Again, this is a nice example of how a kid works through some advanced mathematical ideas.

Next we went to the computer to begin looking at the last two digits of Graham’s number. The last two digits of powers of 3 repeat every 20 powers, so it was easier to use Mathematica to find the cycle than it was to do it by hand.

Here I just explain the short little computer program I wrote to him.

Finally, we tried to see if we could use the idea that the powers of 3 repeat their last two digits every 20 steps to see if we could find the last 2 digits of Graham’s number.

As we started down the path here, I didn’t know if we’d find those last two digits. But we did! It was a nice way to end the project.

The square problem from the Julia Robinson math festival part 2

Yesterday we did a fun project on a problem I learned from Michael Pershan

That project is here:

Sharing a problem from the Julia Robinson math festival with the boys

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:

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.

Sharing a problem from the Julia Robinson math festival with the boys

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:

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.

Working through a neat problem from Martin Weissman’s An Illustrated Theory of Numbers

I just got back from a wo7rk trip to Sydney and I’m going to blame jet lag for goofing up the videos. Because I forgot to zoom out after zooming in during the first video, this is really more of an audio project than a video one!

Today we returned to Martin Weissman’s An Illustrated Theory of Numbers. Flipping through the chapter on prime numbers (which is incredible!) I ran across a problem dealing with the set of numbers {1, 4, 7, 10, 13, \ldots } and thought it would be a great one to talk through with the boys.

It was really fun as you will see hear . . .

I stared by introducing the problem and also making it impossible to see what we were doing:

Next we started playing with the first part of the problem. What we talk through here is this idea from number theory: If two numbers A and B are in our set, and A = B*C, then C is also in the set.

The boys looked at a few examples initially and noticed that lots of numbers in the set didn’t factor in the set. Then they noticed that the problem was really a problem about modular arithmetic.

The next part of the problem we played with was going through an exercise similar to the “Sieve of Eratosthenes” procedure to find the “primes” in our set:

Finally, we took at look at the part of the problem that caught my attention -> find elements of our set that factor into irreducible elements in non-unique ways.

My older son found one example -> 100 = 10*10 = 25*4.

The property of our set shows that the integers factoring into primes in a unique way is actually a pretty special property.

Sorry for the filming screw up – fortunately the visuals for this project were quite a bit less important than average. I’m excited to play around in the project chapter this week – I really love this book!

How a kid approaches a challenging problem

We stumbled on this problem in the book my older son is studying over the summer:

A game involves flipping a fair coin up to 10 times. For each “head” you get 1 point, but if you ever get two “tails” in a row the game ends and you get no points.

(i) What is the probability of finishing the game with a positive score?

(ii) What is the expected win when you play this game?

The problem gave my son some trouble. It took a few days for us to get to working through the problem as a project, but we finally talked through it last night.

Here’s how the conversation went:

(1) First I introduced the problem and my son talked about what he knew. There is a mistake in this part of the project that carries all the way through until the end. The number of winning sequences with 5 “heads” is 6 rather than 2. Sorry for not catching this mistake live.

(2) Next we tried to tackle the part where my son was stuck. His thinking here is a great example of how a kid struggling with a tough math problem thinks.

(3) Now that we made progress on one of the tough cases, we tackled the other two:

(4) Now that we had all of the cases worked out, we moved on to trying to answer the original questions in the problem. He got a little stuck for a minute here, but was able to work through the difficulty. This part, too, is a nice example about how a kid thinks through a tough math problem.

(5) Now we wrote a little Mathematica program to check our answers. We noticed that we were slightly off and found the mistake in the 5 heads case after this video.

I really like this problem. There’s even a secret way that the Fibonacci numbers are hiding in it. I haven’t shown that solution to my son yet, though.

Revisiting Graham’s number

We had only a short time this morning for a project. At this point I should know better than to rush things, but I don’t!

Based on some twitter conversations this week I thought it would be fun to revisit talking about Graham’s number. We’ve done several projects on Graham’s number in the past, but not in at least a few years.

To get started, I asked the boys what they remembered:

Next we talked about one of the simple properties of Graham’s number (and power towers) -> they get big really quickly!

Here we talked about why 3^3^3^3^3 is already as about as large as one of the usual “large” properties listed for Graham’s number. Namely, you couldn’t write down all the digits of this number if you put 1 digit in each Planck volume of the universe:

Next we talked through how to find the last couple of digits of Graham’s number. This part of the project is something that I thought would go quickly, but didn’t at all. Still, it is pretty amazing that you can find the last few digits even though there’s next to nothing you can say about Graham’s number.

If you search for “Graham’s Number” in my blog or in google, you’ll find some other ideas that are fun to explore with kids. I highly recommend Evelyn Lamb’s article, too:

Evelyn Lamb’s amazing article about Graham’s number

A quick look at remainders

My older son was learning about the polynomial remainder theorem yesterday and then the Theorem of the Day twitter account tweeted about the theorem:

I took it as a sign that we should review remainders. My younger son doesn’t have a lot of experience with polynomials, so I wanted the main focus of today’s project to be on remainders when dividing integers. Here’s how we got started:

Next we looked at remainders in different bases to see what was the same and what was different:

Now we looked at the relationship between divisibility rules and remainders

Two wrap up, we looked at polynomials. Obviously this part is not meant to be comprehensive as my younger son isn’t that familiar with polynomials. What I was trying to do here was just give a simple overview of the remainder theorem for polynomials, and show that it wasn’t really that different than what we’d just looked at for numbers.

It was definitely a fun surprise to see the polynomial remainder theorem show up in two totally different places yesterday. Hopefully this review of remainders today was a nice exercise for the kids and helped my older son see a connection between division with integers and division with polynomials.