Counting and a fun harmonic series fact

Saw this tweet (and embeded comic) from Evelyn Lamb last week:

I remembered the comic this morning the boys and I were talking about section 2.3 of Art of Problem Solving’s Introduction to Counting and Probablilty – complimentary counting. Though the divergence of the harmonic series wasn’t something that they were likely to remember, showing them why most large integers had the digit 9 seemed like a fun complimentary counting project.

First I just showed them the problem and we had a fun little discussion about infinity. They didn’t remember seeing the harmonic series before, but, of course, they remember the Numberphile 1 + 2 + 3 + . . . = -1/12 video 🙂


After that quick introduction, we moved on to discussing why most integers will have 9 as one of their digits. In this part we talk about 1 and 2 digit integers as well as introducing a way to count the numbers with 9 as a digit using complimentary counting.


Next we moved on to 3 digit numbers. The boys counted the number of three digit numbers with 9 as a digit using complimentary counting. After that I showed them how you could count all of the number from 0 to 999 with at least one 9 all at once:


Now we moved on to six digit numbers. The original goal was to see if more than half of the six digit numbers had 9’s. Almost!!

In the middle of the video my son wondered why the number of integers with a digit of 9 kept increasing. I tried to give him a quick explanation. This part also provided a sneaky way to talk about / review powers.


For the last part of this project I wanted to explain in a more intuitive way why it makes sense that almost all really numbers have a digit of 9. For this part we talked a little bit about basic probability. I’m not sure if this approach was the right way to explain this idea to kids – my younger son was confused about adding vs multiplying probabilities for example – but it was fun to try. Also some neat ideas about infinity right at the end 🙂


So a nice little project inspired by a fun little math comic from last week. Quite a happy coincidence that the problem had a little bit of a relation to the counting tecnique we were talking about today.

Geometry and counting

We started a new chapter in our Introduction to Counting and Probability book today. The chapter covers “basic counting techniques” and is filled with wonderful counting problems. We worked through one of the examples from section 2.2 – “Casework” – for our project today.

The problem is simple to state, but coming up with all of the cases requires some careful thinking:

How many squares of any size can be formed by connecting the dots in a 5×5 grid?

I thought using snap cubes for the grid would help the boys understand the problem (though I’m not sure if it helped that much). Here’s what the table looked like at the end of the project:

Counting Pic

We started by identifying the cases. First the cases where the sides of the squares are parallel to the edges of the grid:


Next the cases where the sides of the squares aren’t parallel to the edges of the grid. The first two parts here were actauly one continuous discussion. I thought breaking the discussion into two pieces made sense for this writeup. Finding all of the different squares in the 2nd part is challenging.


Next we moved on to counting the number of different types of squares. The first thing we did was count the squares whose sides were parallel to the grid. It took a minute for the boys to understand the geometry in this situation, but once they made sense of the geometry for the first case, all four cases went fairly smoothly:


Now things get a little tricky! Counting the number of squares whose sides are not parallel to the grid. There are two different types of squares here – the “\sqrt{2}” squares and the “\sqrt{5}” squares, and there’s an interesting difference in the way these figures interact with the grid.

First the \sqrt{2} squares:


Next the \sqrt{5} squares. Counting these cases helps you to understand a bit about rotations, and we used some extra snap cubes to illustrate the rotations for the camera:


So, a really neat counting problem. It is fun to see the connections between geometry and counting in this problem. Another fun piece of this project is that there’s not a lot of math background needed to tackle this problem. Hopefully that makes it a great introductory counting problem for kids.

Happy Tau day!

Since the kids are just back from a week of camping, and consequently a little tired, I thought we’d do a fun Tau day project. Turned out that we got a little sidetracked on a little geometry point. Still a fun project, though, just not what I was expecting.

We started with this old Numberphile video about \pi and \tau:


After we watched the Numberphile video, we began our conversation about \tau. One point I wanted to focus on for a bit – and I really thought it would be about a 1 minute thing – was Matt Parker’s point that the diameter was easy to measure. The boys didn’t remember this point from the Numberphile video, and talking about how to measure the diameter started us down a long path:


We made a circle with our compass off camera to help us explore the question of how to find the diameter. My older son had the interesting idea of drawing a square around the circle. If you could do that, then finding the diameter would be pretty easy. The trouble is – how do you draw that square?

At the end of the video, my younger son suggests that we measure the circumference to find the diameter.


Now, following the suggestion at the end of the last video, we found some string and tried to measure the circumference. We found the circumference was about 12.5 inches. That measurement led to a long discussion about how to calculate the approximate radius if we knew the length of the circumference.


Following the discussion about the circumference, we returned to trying to measure the diameter directly. This measurement problem really gave the boys fits. Part of the confusion, I think, was that they were looking for a way to find the diameter exactly. There are, of course, plenty of ways to do that, but looking for the absolute perfect solution was distracting them from using the ruler to find a close approximation.

At the end of this video we stumble on an important idea – the diameter is the longest line segment that you can draw in the circle!


The idea at the end of the last video gives as a way to get an approximate measure of a circle’s diameter – we just look for the longest line that we can draw. Both kids had some interesting ideas about how the length of lines would shrink or grow as you moved around the circle. Exploring those ideas allowed us to get better and better approximations for the diameter. Hopefully the shadows don’t obscure the measurements we are making in the video.


So, although the project didn’t quite go in the direction that I was expecting, a pretty interesting project. It is nice to see that an offhand comment from a mathematician – in this case that the diameter of a circle is easy to measure – can lead to a fun little project for kids.

A challenging Venn diagram problem

[note: written up super fast without any editing]

We did this project with the boys last week before they went to camp for a week. Unfortunately I never got around to writing it up – here it is fairly quickly.

First we talked through the problem (again, sorry for the camera focus at the beginning – lasts 10 seconds). I wanted to go slowly introducing the problem because it is easy to go wrong right at the beginning in these problems:


In the second part we start in on the solution of the problem. My younger son takes the lead by finding how many students take only calculus. After that we spend a few minutes making sure that we know how to count the students in the various different buckets. They adopt a strategy of counting what we know and then using that information to help us find what we don’t know. By the end of this video we have found that we have 105 people to fit into the two open buckets, and we have an expression for one of those empty buckets in terms of x.


We start off this section by finding an expression for the number of people in our last remaining buck in terms of x. We now add our two expressions involving x together to get an expression. From the last video we know that this expression has to be equal to 105. After a quick review of how we got all of this information, we find that the number of people taking physics is 110:


The last part of the project is one final review of the whole problem. I wanted to go through it one more time just to double check that both kids were able to follow all of the steps:


So maybe not our best project ever, but it was nice to see that the kids were comfortable solving a challenging problem like this one. There’s lots of information to keep track of here and I’m glad they were able to see this one to the end. After their week at camp, we’ll start in on the next section of our Introduction to Counting and Probability book tomorrow. That section is about basic counting techniques.

Cathy O’Neil on Trig

Cathy O’Neil published this awesome piece about trig on her blog yesterday:

Fuck Trigonometry

It created quite a conversation. Yay!

Her husband’s comment at the end of the post caught my attention and I spent most of yesterday sort of daydreaming about his point:

When I mentioned my hatred of trigonometry to my husband, he countered with an argument that wasn’t mentioned so far. Namely, that we have really no reason to teach high school kids any given thing, so we just choose a bunch of things kind of at random. Moreover, he suggested, if we remove trig, then meeting people at an airport would just elicit some other reason for hating math. We’d be simply replacing trig with some other crappy topic choice.

I think I’m leaning towards agreeing with him. I’m certainly not sure I could make a convincing argument about why trig *needs* to be taught. In fact, with 3D printing and maybe even Zometool sets becoming cheaper and easier to find, my vote would probably be to try more fun geometry projects before diving into trig. Our Gosper curve project, for example, is something that I think kids would find more entertaining that trig identities:

Exploring the Gosper Curve

The passion in the conversation around Cathy’s post also surprised me a little – I didn’t realize that so many people had such strong feelings about trig! Most of the internet math flame wars I see are about addition or fractions – watching people fight about trig was so refreshing 🙂

Thinking back to my high school trig class with Mrs. Kovaric yesterday I honestly couldn’t remember really having any strong feelings one way or another. Without any strong opinions to fight about (ha ha) I started thinking about some fun math ideas related to trig that I’d learned either in high school or in college. Not reasons to teach trig, for sure, but definitely more fun than memorizing identities!

(1) The Extended law of Sines

One idea everyone sees in trig class is the law of sines – in any triangle ABC, A / Sin(A) = B / Sin(B) = C / Sin(C). Pretty neat relationship, but if these three expressions are all equal to each other is their value special? Turns out that it is:


(2) Stewart’s theorem

This is a cool theorem which gives the length of a line segment from a vertex of a triangle to the opposite side. As with the extended law of sines, this theorem is something that I found in Geometry Revisited in high school. The proof (that I know) involves the law of cosines:


Also, the law of cosines came up in a surprising way in an introductory geometry talk I had with my older son this past school year. This conversation was an unexpected (to me) way that you could talk about the ideas behind the law of cosines in geometry class:

When we accidentally derived the law of cosines

(3) The sum of the inverse squares

Using the Taylor series for Sin(x) and the fact that the roots are integer multiples of \pi, you can prove that:

1 + \frac{1}{4} + \frac{1}{9} + \ldots + \frac{1}{n^2} + \ldots = \frac{\pi^2}{6}

It was incredibly cool to learn that there was a known formula for all of the inverse even powers (solved by Euler in the 1700s, if I remember right), but that a closed form for the odd powers greater than 1 was not known. This is a neat example of an unsolved math problem that high school students can understand and even play around with a little. I’ve always hoped to see a closed form solution for the sum of the inverse cubes.

Another fairly famous trig-related sum problem that blew me away in high school is this incredible sum:

Let x_n be the n^{th} positive solution to the equation x = Tan(x). Find \sum \frac{1}{(x_{n})^{2}}

The particularly amazing thing about this problem is that you can find the sum even though you can’t write down a closed form for any of the expressions that are in the sum!

(4) A surprising integral

I went to college planning on majoring in aerospace engineering – that’s what you do with math, right 🙂

Sitting in an introduction to complex analysis class my freshman year, I ran across this interesting little problem:


Seeing this problem made me want to major in math rather than engineering – it was absolutely amazing to me that \pi and e could be connected in such a seemingly mysterious way.

(5) Circles on a sphere

This one is the one and only time that I’ve used trig directly at work (probably more than 10 years ago, though I don’t remember the exact timing).

One of the guys in our office who thinks about hurricane insurance had a list which gave the latitude and longitude of the center of every hurricane that hit North America for the last 50 or so years. The list had coordinates for the center in time increments of 6 hours. The question he wanted to answer was relatively simple: given a specific latitude / longitude (say Miami or New York City, or something) how many Hurricanes had come within a given distance of that city (50 miles, 100 miles, . . . .).

He’d tried to write a really quick and back of the envelope program to answer this question but it was giving answers that seemed really wrong. To calculate distance correctly you need a little bit of trig because you have to factor in how far north you are. Adjusting the distance formula for a given latitude helped him get to the right answer. There were a few other little math-related tricks in the program, too, such as checking whether or not the path between two points came within the desired distance even if the endpoints were outside of the distance. Without trig, the distance calculations in this project were easy to get wrong.

Anyway, not a list of reasons to teach trig, but rather just a few fun trig-related things that Cathy’s post got me thinking about. Hopefully slightly more fun than memorizing identities 🙂

Although if you’ve made it this far and do like trig identities, though, a recent Terry Tao post should be right up your alley:

A “cute” differentiation identity

Odds of winning the US Open

Since we are doing a summer project on counting and probability I thought it would be neat to talk a little bit about odds in sports. We decided to look at the odds for a few of the players to win the US Open golf tournament for today’s project.

I grabbed the posted odds from the European sports aggregator site Oddschecker and asked the kids if they understood what the odds meant. I wasn’t expecting them to understand odds completely, but they did have some interesting ideas:


After a brief introduction to odds, we took a quick look at what it meant that Jordan Spieth had odds of 2 to 1 to win today. My youngest son thought these odds meant he would win half of the time. My oldest son was a little confused by how sports betting worked, but I tried to simplify the ideas by just talking about the amount of money you start with and the amount you end with.


After working through Jordan Spieth’s odds of winning, we tried to understand the odds for a few of the other players. Dustin Johnson, for example, had odds of 11 to 4 – a tiny bit harder to understand than 2 to 1.


Now we switched to looking at the odds of finishing in the top 5. One of the nice things about Oddschecker is that you can find odds for lots of unusual bets. The listed odds for Jordan Spieth finishing in the top 5 today were 1 to 5. I wanted to look at this case since the high and low numbers are reversed from the first example that we looked through.

This section went a little long because of a little confusion about what 1/6th is as a percent. I’m not sure where the confusion came from, but we spent the last 4 minutes of the video sorting out how to write 1/6th as a percent.


We started this last section by finishing up the talk about 1/6th and then moved on to talking about both the odds and percentages of the other players finishing in the top 5. Here we encountered some more fractions that we needed to convert to percentages and the work we did on 1/6th seemed to help.


So a fun project that is somewhat related to the summer project that I’m working through with the boys. Hopefully next time they hear about odds they’ll have a little bit better understanding of how they work. Also, I’m happy that we ran across some of the confusion about fractions and percents. It has been a while since I covered percents with them and the struggles today gave me a nice reminder to spend a little time reviewing percents at some point this summer.

Counting, Pascal’s triangle, and binary numbers

In yesterday’s counting project the boys noticed a connection between a counting problem and binary numbers. Here’s that project:

Revisiting an AMC 10 Counting Problem

For today’s project I wanted to explore that connection an a little more depth. To start off we looked at the connection between counting arrangements and Pascal’s triangle:

In the first part of this project we saw a connection between counting pairings of tourists and guides has a interesting connection with Pascal’s triangle. Here we look more carefully at that connection by trying to understand how the rule that tells you how to construct the rows of Pascal’s triangle also shows up when you count these pairings.

We explored the connection here in two parts. In this first part we show the 6 ways that you can pair 4 tourists with 2 guides when each guide has 2 tourists. We also show the 3 ways to pair 3 people with 2 guides where the first guide gets 1 person and the 3 ways to pair 3 people with 2 guides where the first guide gets 2 people.

Now we are ready to find the connection between the two lists me made in the prior video. That connection is important because it shows that the same addition rule that gives the rows of Pascal’s triangle also applies to counting arrangements of certain sets, and therefore helps you understand why Pascal’s triangle helps you count those arrangements.

In the last part of the project we explore the connection between binary numbers and Pascal’s triangle. We do this using an example of 5 digit binary numbers (from 00000 to 11111). This connection allows you to see that the rows of Pascal’s triangle always add up to be a power of 2.

So, a nice little project showing some fun connections between Pascal’s triangle, counting, and binary numbers. Some of these connections are pretty deep and I certainly don’t expect that the boys will have understood every detail from this project. They did seem to have fun with it, though, and their understanding seems to have come a long way from when we worked through the AMC 10 problem earlier this week.

Revisiting the old AMC 10 Counting problem

Yesterday we struggled through an old AMC 10 counting problem:

A bit of a struggle with our counting project today

Here’s the actual problem:

Problem 12 from the 2007 AMC 10a

Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?

After talking a little bit about permutations and combinations this morning, I thought we’d give the problem a second try. Not surprisingly it was a little easier the second time through, though there were still some challenges. We ended by talking about how this particular problem is related to counting in binary.

First, though, a quick review of permutations and combinations:

After that quick review we jump into the AMC 10 problem. Our first approach involves counting the ways that 1 person can go with the first guide, then 2 people with the first guide, and so on. My younger son got a little confused about the case with 2 people, so we looked at the case extra carefully. He was confused about whether or not order mattered.

With the confusion about order (hopefully) sorted out, we continued counting through the remaining cases. Here we also have a nice chance to talk about some of the symmetry in this problem.

Next we looked at the problem from a different perspective – the perspective of the tourists. This approach allows us to get to the answer to this problem in a way that seems completely different that the first approach. One fun thing my older son noticed here is that it seems as though our approach is related to counting in binary.

Finally, we looked more carefully at the connection to counting in binary. This connection probably deserves a project all to itself, but even going through things quickly here was fun.

It felt as though this project went much better than yesterday’s project. People naturally snicker a little bit when you say it, but learning to count is really hard. Yesterday the confusion involved adding and multiplying. Today the confusion involved whether or not order mattered. Working through those two ideas isn’t necessarily super easy, but I think we made some good progress. Fun little project.

A bit of a struggle with our counting project today

My older son and I were looking at this problem the other day:

Problem 12 from the 2007 AMC 10a

Here’s the problem:

Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible?

Since we are studying counting, I thought reviewing this problem would make for a good project with the boys this morning. Unfortunately the project didn’t go as well as I was hoping, but we did get to have a good conversation about the times when you need to add and the times that you need to multiply when you count.

We started the project by having my older son explain his original solution. His approach is fairly straightforward and gave a couple of different opportunities for my younger son to help out. I’m glad that my younger son was able to participate because there are some fairly challenging counting ideas here:

We ended the last video with my younger son being a little confused about whether or not to add or multiply when we’ve looked at some cases. I decided to take a look at a slightly easier problem so that we could actually list out all of the cases. I hoped this exercise would help him see that we needed to add in the last video. Unfortunately writing out the cases was a little harder than I thought it was going to be.

However, there’s some pretty good ideas from both kids right at the end about the relationship between the numbers we have on the board and Pascal’s triangle.

For the last part of the project I showed them a different way to approach the problem – looking at it from the perspective of the tourists. We had a little more confusion about adding and multiplying. I tried to clarify that we needed to multiply here, but I’m not sure my explanation helped.

So, a fun problem, but not such a great project. Can’t win them all 😦

Responding to Dan Meyer’s Quadratic question

Interesting question in Dan Meyer’s blog today:

Dan Meyer’s blog post from June 17, 2015

Here’s the question:

Instead, ask yourself, “Why did mathematicians think this skill [ factoring quadratics with integer roots ] was worth even a little bit of our time? If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?”

I happened to see this post right before leaving the office. On my 45-ish minute bike ride home I spent some time thinking about why I thought this skill was worth some time. Here’s where my thoughts took me . . . .

The first thing I thought of was a passage from Jordan Ellenberg’s How not to be Wrong. On page 323 in the section “The unreasonable effectiveness of classical geometry” he makes this point about why ellipses seem to show up all over the place in math:

In math there are many, many complicated objects, but only a few simple ones. So if you have a problem whose solution admits a simple mathematical description, there are only a few possibilities for the solution. The simpleset mathematical entities are thus ubiquitous, forced into multiple duty as solutions to all kinds of scientific problems.

I think quadratic equations fall into the bucket of simple mathematical entities that Ellenberg is talking about here, and I’m not surprised that math folks would think they are worth of study. The simplest case is probably quadratics where the solutions are integers, so that’s a natural place to start.

But there are other important mathematical ideas that you see – maybe for the first time – when you start to think about factoring quadratic equations. Here are three ideas that I thought of specifically:

(1) Factoring integers

Mike Sipser, the former head of MIT’s math department, has a nice public lecture which includes a discussion of difficulty of factoring integers in the first 10 minutes:


By the time a student encounters quadratic equations, he or she will have had a lot of practice multiplying numbers but probably less practice factoring them. When factoring x^2 + bx + c you have to grapple with the problem of finding two numbers that multiply to be c and sum to be -b. As Sipser’s lecture shows, this is by no means an easy question – particularly when you are seeing it for the first time.

I asked my kids to take a crack at finding two integers whose product was 120 and whose sum was 26 as an illustration:



One more advanced project that I have done with my older son that involved factoring polynomials was based on a neat post by University of Colorado math professor Richard Green that Patrick Honner pointed out on Twitter:

A “new to me” proof that there are infinitely many primes

(2) Exploring properties of numbers in depth

Prior to encountering quadratic equations, students will (hopefully!) have studied and solved linear equations like 3x - 6 = 0. My kids, at least, will solve equations like this by moving the 6 to the “other side” and then dividing by 3. Great technique for linear equations, but not so great for quadratics. You need a new idea and that idea is pretty deep – if two numbers multiply to be 0, then one (or both) of the numbers has to be zero.

You’ll can hear Julie Rehmeyer talk about struggling with a similar idea in this Inspired by Math interview. The part I’m referencing begins around 31:30 when she talks about her time at Wellesley and trying to prove that 0 + 0 = 0:

July Rehmeyer interviewed by Inspired by Math

One other pretty profound idea that you encounter for the first time with quadratics is finding that equations can have multiple solutions. My 3x - 6 = 0 example has the solution x = 2. With quadratics you can see 2 solutions – but what does that even mean? It has to be very confusing seeing multiple solutions for the first time. Sometimes those second solutions are fun to explore, though:

Dan Meyer’s Geometry Problem

Also, going beyond two solutions can be interesting to kids:

Quadratic and Cubic equations

(3) Finding new types of numbers

This part isn’t so much about factoring as it is about quadratic equations in general. Or maybe just quadratic equations with integer coefficients rather than ones that factor into integers. In any case, with some simple quadratic equations that will not factor easily you are able to talk about some new kinds of numbers:

I saw two interesting pieces from prominent mathematicians talking generally about numbers. The first piece was Ed Frenkel’s book Love and Math. I used some of Frenkel’s ideas about quadratic equations to talk about some surprising similarities between \sqrt{2} and i with the boys:

Ed Frenkel, the square root of 2, and i

The next piece was the public lecture that Jacob Lurie’s gave after winning the Breakthrough Prize. His lecture is an absolutely wonderful talk about math from a mathematicians point of view, and it has a couple of great ideas that you can use with kids. In the first three minutes of the lecture you can see some of the important ideas about irrational and imaginary numbers that come into play with quadratic equations:

Using Jacob Lurie’s Breakthrough Prize lecture with kids

All of this was a long way of saying that I think quadratic equations serve as a gateway to some interesting and advanced mathematical ideas as illustrated in the three points above. They also come up in enough places in math and physics (hinting at Ellenberg’s idea) that I’m not really surprised to see that mathematicians think they are important to study.

Though a subjective feeling, obviously, I feel that kids will find many of the ideas related to quadratic equations to be fascinating, which is why I’ve tried them out with my own kids.

Oh, and just as I was finishing writing up this post I remembered another fun project with the boys that involved a quadratic that can factor over the integers – probably the most internet famous math problem so far in 2015 🙂

Talking with the boys about Hannah and her sweets

So, I guess that’s sort of the complete set of thoughts from the bike ride home 🙂