A triangle problem from Cliff Pickover

Saw this problem on twitter this morning:

When I asked what the kids wanted to talk about today, my younger son said that he wanted to talk more about my the “things you should know” list that the math team from my older son’s school.

Well, that list had several ideas about areas of triangles, so I used the tweet as a starting point.

Here’s what each kid thought was the answer plus the calculation of the area of the first triangle:

In the last video my older son mentioned that we could also use Heron’s formula to calculate the area of the triangle. My younger son hasn’t seen that formula, yet, but I thought it would be fun to show him that formula:

Next we move to the other triangle and find the answer to the question in the Pickover tweet:

Finally, I thought it would be fun to look at Heron’s formula for a triangle whose area wasn’t so easy to calculate using other methods. I asked the boys to choose a random triangle and my younger son chose three sides that do not actually form a triangle (of course!).

That caused a fun little diversion at the beginning of this video, but once we worked through the area of that triangle we studied an actual triangle with Heron’s formula.

So, a quick little project using some triangle area formulas. Glad I saw the tweet from Cliff Pickover – it made motivating this project really easy.

Walking down the path to the surreal numbers part 2

I’ve got less time to write today because of a family trip, but the videos below show part 2 of our Family Math project about “checker stacks” and the surreal numbers.

The first part of the project is here:

Walking down the path to the surreal numbers

and we are following Jim Propp’s blog post about the surreal numbers which is here:

Jim Propp’s “The Life of Games”

The first thing we looked at today was “deep blue” stack. The surprise about this piece in the game of checker stacks is that its value appears to be positive infinity.

Next we quickly looked at the “deep red” piece and then looked at a blue + deep red stack whose value is pretty surprising. It was great to hear the ideas that the kids had about this stack.

Next we moved on to study the “deep purple” piece. This piece is pretty mysterious. I thought pretty hard about how to explain the value of this piece to the boys, but didn’t really come up with any good ideas. Instead we spent about 10 minutes exploring its value. That was a great conversation, but we never did quite get to the value of 2/3 that Propp gives in his blog. I’m ok with that outcome, though – I felt the conversation about the possible values was really great.

So, sorry for the quick write up of this 2nd project about the surreal numbers. I’m really happy to have seen Jim Propp’s blog and think there’s got to be a great way to use checker stacks for a neat math project for kids.

The NY Times’s 8th grade math questions

Saw this tweet earlier today from Daina Taimina:

A direct link to the NYT article is here:

8th grade math questions in the NYT

I’m terrible at determining what sorts of questions will be difficult for kids, so I asked my older son (who is going into 6th grade) to talk through them. Here are his responses – the last two questions gave him a little difficulty.

(1) An algebra / arithmetic question – his solution is actually pretty clever

(2) A geometry / angle question – here he makes a little arithmetic mistake, but luckily the answer he finds after this mistake isn’t one of the choices.

(3) A question about similar triangles – this one also has a bit of arithmetic, but it all goes well.

(4) Some 3D geometry plus arithmetic with decimals. This one gives him trouble on two fronts. He uses an incorrect formula for the volume of the cone and has a bit of difficulty estimating the product (I don’t know if calculators were allowed on the exam or not, but we were just standing by the computer).

It was interesting to see the struggle estimating the product.

(5) More 3D geometry and arithmetic with decimals. Here he also had a little bit of trouble estimating the product he needed to compute. Luckily, though, the answers were far enough apart that even with that difficulty he was able to identify the likely answer.

I was surprised to see that this question – which seemed to me to be a little easier than the previous one – only had a 49% correct response rate compared to 70% on the question with the cone. Interestingly, right at this moment, those numbers are almost exactly reversed in the reader answers.

Walking down the path to the surreal numbers with kids

For reasons that are somewhat mysterious to me, I’ve seen a bunch of media coverage about John Conway in the last couple of months.

Of course, there’s Siobhan Roberts’s awesome biography:

Which Jordan Ellenberg reviews in the Wall Street Journal here:

Jordan Ellenberg’s review of Genius at Play

Roberts also has a great Quanta Magazine article:

John Conway: A life in games

One other item that caught my attention last week was a Jim Propp blog post I saw tweeted out by Jordan Ellenberg:

Jim Propp’s “The Life of Games”

Propp actually wrote some of my graduate school recommendation letters back in . . . oh why did I bring this up . . . 1992, so it was cool to learn that he had a math blog!

Anyway, Propp’s piece describes the game of checker stacks so well and so simply that I thought it would be fun to try to talk through some of the ideas with the boys. We didn’t get to some of the more unusual features of the game today – and so didn’t really get to the surreal numbers – but even showing the boys how the number 1/2 shows up in the game was really fun. We’ll take a few more steps down the path to the surreal numbers tomorrow.

Here’s how today’s project went (oh, and sorry that the camera angle is so bad in the last two videos, I didn’t notice that the tripod got bumped until I was publishing the videos.):

The first step was a quick introduction the game of checker stacks and a few thoughts from the boys about the game:

Next we studied what happens when we stack checkers on top of each other. First we studied the relatively simple situation – a game with a single red vs. a stack with two blues. Then we moved on to a more complicated situation of a single red vs. a red blue stack. The boys were able to get their arms around these two situations, which was nice.

I realized at the end of the last movie that I wasn’t following Propp’s presentation correctly – he was using blue red stacks rather than red blue stacks. The difference becomes important when you are trying to find a game when it matters which color goes first. So for the third movie we we now studied the blue red stacks.

Also, the boys remembered in between these two videos that we had some 3d-printed red and blue action figures!

One interesting bit of math from the kids in this video was that they assumed that all of the stacks would have values that were represented by integers. Because of that assumption, they think the value of the blue red stack must be 0 or lower. We’ll explore that idea a little more in the last video.

The last game we study is a game with a single red vs two blue red stacks. Propp uses this game to illustrate a surprising property of the blue red stack. The boys were indeed surprised by the result ðŸ™‚

So, a really fun start to our journey to the surreal numbers. It was neat to see that even some fairly simple positions from the checker stacks game gave the kids some challenging things to think about – and even a few surprises!

The sum of the divisors of an integer

Today we can to the end of our short project on divisors. The topic for today was finding a formula for the sum of the divisors of an integer.

I wasn’t entirely sure how to go about explaining this idea to the boys – for one thing a thorough explanation sort of requires some geometric series formulas – so I just let them take the lead.

We started by reviewing the formula for the product of the divisors and then began to talk about the sum. I posed the question about the sum as a challenging math problem with the specific idea of thinking about how to break this problem down in to smaller pieces that we could understand.

In the last video we found formulas for the sum of divisors of a prime and the square of a prime. We kept going with that idea at the beginning of this video and then the boys started noticing a pattern:

At the end of the last video we started forming an idea about the formula for the sum of the divisors of a number. We spent another 5 minutes exploring that formula before moving to Mathematica:

Finally, we went to Mathematica to play around with other numbers a little more quickly.

So a fun week with a little number theory and a little computer math. The project this week were sort of inspired by a “things you should know list” from my older son’s school math team. Although I was a bit surprised by some of the topics on that list, talking through a few of those topics this week was really fun.

A nice problem for kids from James Cleveland

Saw this problem posted on twitter by James Cleveland:

I thought it would make a nice problem for the boys to talk through. It was actually much better than I expected from the point of view of hearing kids talk about math.

So, without much comment, here are their thoughts. My 6th grader first and then my 4th grader. Probably the most difficult piece for both kids was figuring out that the two solutions they found were, in fact, the only two solutions.

What math is like for me sometimes

This morning the boys and I did a really fun computer / number theory project based on a problem that Dan Anderson sent to us. Here’s the write up of that project:

A neat problem Dan Anderson shared with us

and here’s the original problem, which is problem #12 in Project Euler:

What is the first triangular number with more than 500 divisors?

We solved that problem using Mathematica and also talked about a natural extension of that problem – what is the first triangular number with exactly 500 divisors?

The interesting thing about this second question is that all of the triangular numbers that we found with exactly 500 divisors were even (and in, fact, multiples of 16). After we turned off the camera, we wondered why that was the case.

It was easy to modify the code to look for triangular numbers with exactly 500 divisors that were also odd. We didn’t find any, though – even after searching through the first 200,000,000 triangular numbers.

Why, though? Why would a triangular number with exactly 500 divisors have to be even?

Eventually I realized that I could modify the code even more to make the search more efficient (and the code more and more completely impossible for anyone but me to understand . . . .). Those modifications and efficiencies came from thinking about the math behind the problem.

The math says that a number with exactly 500 divisors has some special properties. For example, it might be a product of 3 primes to the 4th power and 2 primes to the 1st power. It might be a product of 2 primes to the 9th power and one prime to the 4th power. There are a few other possibilities, but not many more. I figured I could speed up the process by searching through the triangular numbers that were divisible by 3^4. This search was still fairly slow, though, and didn’t find anything up into the 300,000,000th triangular number.

What about 5^4? nope – still too slow.

But, I got ahead of myself, because searching for triangular numbers divisible by 11^4 produces these results:

The 484,983,125th triangular number – namely 117,604,316,009,874,375 – is what I now think is the smallest odd triangular number with exactly 500 divisors.

Yay!!

But if you look carefully, you’ll see something else in that picture. Something that only appeared because I wasn’t at the computer to stop it after the other discovery. Namely, the 1,482,401,249th and 1,482,401,250th triangular numbers both have 500 divisors and are both odd! Twins!!

The numbers are enormous and their factors are really cool:

$1,098,756,732,259,580,625 = 3^4 * 5^4 * 7^1 * 11^4 * 211,771,607^1$ and

$1,098,756,733,741,981,875 = 3^4 * 5^4 * 11^4 * 3383^1 * 386,747^1$.

I wonder if there are infinitely many of these odd twins??!!

I don’t get to do a lot of math like this any more. A day of playing around, thinking about the math, and a lot of trial and error was really fun and even produced some interesting results. That’s really what math is about for me.

This evening I sat down with the kids to talk about the fun we had today and ask them if there were any other searches they’d like to try out. Here are those two quick talks:

A neat problem Dan Anderson shared with us

We’ve spent the last couple of days studying divisors of integers – mainly the number of divisors and the sum of those divisors. This topic came to us via a “things you should know for math contests” list that math team at my older son’s school gave to the kids.

We’ve used Mathematica to help us get a feel for these topics and that computer work (I assume) prompted Dan to share this problem with us:

The problem is: Find the first triangular number with more than 500 divisors.

I asked the boys if they wanted to try to tackle this problem, and they wanted to give it a try. So . . . off we went:

Once the kids understood the problem, I thought it would be useful to spend some time talking about how we could approach the problem. The boys had some pretty good ideas:

The one thing I wanted to spend some extra time on was an alternate way to calculate the triangular numbers. The method that the boys proposed was actually fine, but it seemed like an extra couple of minutes talking about a different approach would be time well spent:

Now we went to the computer to implement our plan. We found that the 12,375th triangular number, 76,576,500, was the first triangular number with more than 500 divisors:

The boys were a little surprised to learn that the first triangular number with more than 500 divisors was smaller than the one with exactly 500 divisors. In fact, we didn’t even find one with 500 divisors yet. In the next part of the project we looked for that number. We did find that number, but it was much larger than we expected – the 1,569,375th triangular number is 1,231,469,730,000, which has exactly 500 divisors!

We wrapped up by looking at 5 of the triangular numbers with exactly 500 factors. They all shared a common factor of 16. We decided to look to see if there was an odd triangular number with exactly 500 factors. As of now (3 hours) after finishing up the project, the computer has not found one.

So, a really fun computer project with the boys. Thanks to Dan Anderson for providing this challenging problem!

Exploring Sums of Divisors with kids

This project is the second in week long series about some topics that my son’s math club will be studying at school. The topic for today is sums of divisors of a positive integer.

I was a little surprised to see this topic on the list because this is a pretty advanced topic. In fact, in just the short time that we spent talking about the problem this morning we arrived at an unsolved problem! Fun project, though, and we’ll talk more about this topic tomorrow.

I started by having the kids play around on Mathematica to see what they would notice about divisor sums. Some of the things they found:

(1) 9240 has the highest divisor sum out of all the integers from 1 to 10,000,
(2) the divisor sum for primes is one more than the prime, and
(3) the divisor sum for perfect squares is always odd.

Next we spent a little time looking at the divisor sum for the perfect squares. It was interesting to hear why they thought the divisor sum would always be odd:

After the discussion in front of the computer, we went to the white board to look more carefully at the sum of the divisors for perfect squares. We started off this discussion by looking at 36. We also look at 225 to see what happenes in an example where all of the factors are odd:

Now we tried to tackle the general case – why does all perfect squares have an odd number of odd factors?

After that discussion, we returned to the computer to explore when the sum of the divisors of a prime number?

Here’s the start of that discussion:

We finished up this part of the discussion by looking to see how far the pattern we found in the first part would continue. We’ll learn a bit more about this pattern tomorrow:

Finally we looked quickly at the super abundant numbers. These are numbers with unusually large divisor sums. In particular, when you take the sum of the divisors and divide by the number, you get a ratio which is larger than any number below it. Here’s a quick introduction, though my attempt to write a quick program on the fly fails so I had to break this part into two pieces:

So, a fun conversation with the kids today. Excited to dig more into sums of factors tomorrow.

A fun divisor counting project for kids

The math team for my son’s new school has a little brochure telling new students some of the ideas that come up in math contests. I don’t know if this list is something that the kids are supposed to know ahead of time, or a list of things that they’ll be talking about during the year, but I thought it would be fun to spend the week touching on a few of the ideas. Today’s topic fit right in with our summer counting project: counting divisors of a number.

We’ve done a few divisor counting problems in the past, so I hoped this would be an easy review topic for the kids. Here are three of our old projects:

The Same problem two years apart

A neat number theory problem for kids from Tracy Johnston Zager

A neat number theory problem for kids from David Radcliffe

We talked about how to count divisors for maybe 15 minutes and then turned our attention to problem 5.29 from our well-worn copy of Art of Problem Solving’s Introduction to Counting and Probability book.

The first part of the problem is here:

Now that we’ve found the minimum number of primes, what about the maximum:

Next up – what is the smallest positive integer with exactly 20 divisors?y

Last up – the challenge problem: Is there a positive integer smaller than 240 with more than 20 divisors. I was really happy to hear the ideas that the boys had while trying to solve this problem:

After the long discussion on the challenge problem, we went to play around with the question with Mathematica. The kids were incredibly engaged when they saw the various ways we could study the problem (and similar problems) with the computer:

So, a fun project for kids inspired by the math club at my son’s new school and also by Art of Problem Solving’s awesome Introduction to Number Theory book.