For Pi day today we explored the amazing near integer

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 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

The number my older son found was

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 ðŸ™‚

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 factor. I also shows how the numbers of the form 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.

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:

Problem A1 from the 2017 Putnam is a great problem to walk through with kids. There's a bit of number theory and a bit of neat mathematical reasoning. Would be fun to talk through in a group. pic.twitter.com/YsIc01STLI

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.

I didn’t do a very good job managing the time on this project today. The trouble is that there are lots of different directions to go with the ideas and we walked down a lot of different paths.

But – I think this is a great topic to show off the beauty of math and we end with an amazing connection between sums of divisors of integers and .

The topic of sums of divisors of an integer came up in my younger son’s weekend enrichment math program yesterday. I thought it would make for a good topic for a project, so I gave it a go this morning.

The first part of the project was mostly about divisors and the kinds of questions that we could ask about them. A lot of the discussion here is about a question you can ask about the product of a number’s divisors:

Next we began to look at the sum of the divisors of a few different numbers. The boys noticed a few patterns – including a pattern in the powers of 2.

At the end we were looking to see if we could find patterns in the powers of 3.

It was proving to be a little difficult to find the pattern in the powers of 3, but we kept trying. After few ideas that didn’t quite help us write down the pattern, they boys had an idea that got us there.

At the end of this video I showed them that the sum of the divisors of powers of 6 was connected with the sum of the divisors of powers of 2 and powers of 3.

To wrap up I wanted to show some larger patterns in divisor sums, so we moved to Mathematica to play around a bit.

While I was doing the same playing around last night I accidentally stumbled on an amazing fact: As n gets large, the average of the sum of the divisors of the numbers from 1 to n is approximately .

Number theory sure has some fun surprises ðŸ™‚

This is definitely a fun topic and also one that could be used in a variety of ways (arithmetic review, intro to number theory, computer math, . . . ). I wish that I’d presented it better. Probably it needs more than one project to really fit in all of the ideas, though.

The problem shows a neat connection between number theory and geometry -> what is the average number of ways to write an integer as the sum of (exactly) two squares?

We’ve looked this problem previously, but it was so long ago that I’m pretty sure that they boys didn’t remember it:

I started by introducing the problem and then having the boys check the number of ways to write some small integers as the sum of two squares:

In the last video we found that the number 3 couldn’t be written as the sum of two squares. I asked the boys to find some others and they found 11 and 6. My older son then conjectured that numbers of the form couldn’t be written as the sum of two squares. We explored that conjecture.

My son’s conjecture was such an interesting idea that I decided to take a little detour and explore squares mod 4.

Slightly unluckily we were time constrained this morning, so the diversion in the last part left me with a tough choice about how to proceed. I decided to show them a sketch of Gauss’s proof fairly quickly. Don’t know if that was the right decision, but they did find the ideas and result to be amazing!

Even though I had to rush at the end, I’m really happy with how this project went. It is fun to see kids making number theory conjectures! It is also really fun to see gets really excited about amazing results in math!

If you’d like to see another fun (and similar) connection between number theory and geometry, Grant Sanderson did an amazing video about pi and primes:

Here are our the two projects that we did based on that video.

Earlier in the week I saw a tweet announcing a new (and really cool!) result about the Cantor set:

I'm told that the last 2 tweet's link goes to my email (!), so I'll try this: THEOREM (with J S Athreya & J T Tyson): Every x in [0,1] can be written as (u^2)v, where u, v are in the #CantorSet. To appear in @maanow 's #AMMonthly; #arXiv preprint https://t.co/FHUXx1N3Q7

The new result is that any number in the intein the weekrval [0,1] can be written as the product where and are members of the Cantor set.

After reading the paper, I thought that it would be really fun to try to share some of the ideas with kids. The two ideas I wanted to highlight in the project today were (i) the geometric ideas in the construction of the Cantor set, and (ii) the interpretation of the Cantor set in base 3.

I started with a question about base 3 -> how do you write 1/2 in base 3?

Now we looked at constructing the Cantor set by removing intervals. The boys had lots of interesting ideas about what was going on

Next we looked at the incredible property that you can make any number in the interval [0,2] by adding two numbers in the Cantor set. This ideas here were a little harder for my younger son to understand than I was expecting, so I ended up breaking the discussion into two parts.

I think the ideas here are fun for kids to think through – how do I pick a number from one set and a second number (possibly from a different set) to add up to a specific number.

Here’s part 1:

and part 2:

Finally, we took a peek at the result from the paper -> how does multiplication work? This was also a fun discussion. The ideas necessary to see why you can find three numbers from the Cantor set that multiply to any number in [0,1] are obviously way out of reach for kids. However, seeing why the multiplication problem is difficult is within reach.

It is always a real treat to find math that is interesting to mathematicians to share with kids. I think talking through some of the ideas related to this new result about the Cantor set makes for an amazing math project for kids!
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One of the boys had to leave early this morning for a school event, so I was looking for a quick project. With some of the work I did in Mathematica on Taleb’s problem still up on my computer screen, I decided to run through the problem with the boys. The point here wasn’t for them to figure out the solution, but rather to see a neat example of counting techniques used to solve a challenging problem.

I started by explaining the problem and asking them to take a guess at the answer. The boys also had some interesting thoughts about the probability of the balls all ending up in different boxes.

Next we went to Mathematica to walk through my approach to solving the problem. In talking through my approach these ideas from number theory and combinatorics come up:

(1) Partitions of an integer,
(2) Binomial coefficients,
(3) Complimentary counting,
(4) Permutations and combinations, and
(5) Correcting for over counting.

Here’s our quick talk through one solution to Taleb’s problem (and, again, this isn’t intended as a “discovery” exercise, rather we are just walking through my solution) :

To wrap up we returned to the idea of the balls spreading out completely -> a maximum of 1 ball per box. Both boys thought this case was pretty likely and were pretty surprised to find it was less likely than ending up with 3 or more balls in a box!

This problem is little bit on the advanced side for 8th and 6th graders to solve on their own, but they can still understand the ideas in the solution. Also, there are some fun surprises in this problem – the chance of the balls spreading out completely was much lower than they thought, for example – so I think despite being a bit advanced, it is a fun problem to share with kids.

One of my older son’s homework problems asked him to find 3 digit multiples of 7 whose digit sums were also multiples of 7. I was puzzled by this problem had it on my mind most of the day today.

I hoped that talking through it would help me understand what the math idea was behind the problem. Sadly no, but we still had a good talk.

Here’s the problem and the work my son did:

So – still quite puzzled about the problem – I decided to see if there was anything quirky that came up looking at a divisibility rule for 7 with 3 digit numbers. This gave us a nice opportunity to talk about modular arithmetic:

Finally, since I wasn’t making any progress seeing the point of the original problem, I had him talk about other divisibility rules that he knew:

So, a nice conversation, but I’m actually baffled. I’ll have to ask the author of the problem what he was trying to get at – I feel like I’m missing the point.

The second half of the podcast was a really interesting discussion of math education. One thing that caught my attention was comparing math education to music education and the idea of having students do “math recitals.”

Another part that caught my attention was a problem used mainly to see the work of the students rather than the specific answers. That problem is roughly as follows:

Find two numbers that multiply to be 1,000,000 but have the property that neither is a multiple of 10.

Here’s how my younger son approached the problem – it was absolutely fascinating to me to see how he thought about it.

Here’s what my older son did. Much more in line with what I was expecting.

Fun little project – definitely check out the Wrong but Useful podcast if you like hearing about math and math education.