# Semicircle Puzzle

Matt Enlow posted an interesting geometry puzzle on twitter (tweet here), and I think the thing that got me intrigued was his initial challenge: “I can’t tell how hard this problem I just made up is.” Not knowing if there is an elegant/easy/obvious solution or not got me hooked.

I’m going to try to outline my approach/solution, because I sometimes like deconstructing my thinking to see how I actually think/learn… so from this point on… SPOILERS.

Some things that stood out to me… First, it looked like there was initially a single circle in a square, and the circle got cut in half and then it started sliding. So I initially drew the full circle in the square (before sliding), I drew the diagram shown, and then I drew the two semicircles in a rectangle after they fully “slid”… I saw the cut circle “in motion” — but after a short while I didn’t see how that would help me.

Then I drew the image and solved the problem and felt proud about it. But then I realized I drew the picture wrong. I circled the wrong part in my diagram, so you can see. I had the “slice” hit the corner of the rectangles, and then I was able to use similar triangles to come up with a solution.

I was proud but for some reason, probably because Matt’s initial tweet suggested to me that it would be harder than this, something was nagging me about it. So I went back and quickly saw my error. But I have always found that taking a wrong approach can help eliminate pathways to a solution, but might also help me see possible tools to use in a solution. And in fact, this idea of using that “cut line” and similar triangles was important in my pathway to the end.

So when I went back to the drawing board, I wanted to really see how this diagram worked… Some things were fixed (the 12 by 19 rectangle, the fact that the semicircles sort of “slid,” and importantly, the fact that the semi-circles were tangent to the rectangle at two places). So I decided to build this diagram in geogebra (with only one of the semi-circles), and as I built it, I saw that everything hinged on the movable point “G.”

I made the line where the semi-circles touched movable, based on the location of point G. Play around with moving point G here on this web-based geogebra page, and try to get it so the semi-circle on the bottom is tangent to the right and bottom side of the rectangle!

So to me, everything hinged on location of point G, or in other words, the distance from A to G (which is the same as the distance from H to C). We are looking for the location of point G which makes the semi-circle perfectly tangent to right and bottom sides of the rectangle. So to me, those appeared to me as “keys” to the problem. [1]

Sooooo I drew my diagram, and importantly labeled the distance from point A to point G with a variable, a. And then I labeled lots of things in my diagram in terms of that variable and the radius of the semi-circle, r.

I had two variables, so I needed two independent equations. And here is something nice… because I initially went down a wrong path earlier with my mis-drawing, I had already gotten similar triangles in my head! So I got one equation from that.

I hunted and hunted, and found another equation I could get… using the Pythagorean Theorem!

So now I had two equations and two variables.

… and since I knew this was going to be a beast to solve, I just used Desmos, and got that the solution is a=1.5 and r=7.5.

I did a little of the algebraic gymnastics to try to work this out by hand, but it was pretty uninteresting to me and I was pretty convinced that if I really wanted to, I could. To me, getting the equations was the interesting part, and the rest felt like pencil-pushing. So I stopped there. It was nice that the geogebra applet I created seemed to confirm my answer for me:

So that was my process to solving this mathematical puzzle. Who knows – I could also be totally wrong! I’m left thinking of the following:

(1) Is there a more elegant way to come up with the answer? Because the answer is so nice (a diameter of 15?!?!) but it comes out of such an ugly set of equations, I bet there is a nicer way. In other words, is there a better “conceptual” approach that gives a stronger insight into the geometric nature of the setup?
(2) How did Matt come up with this puzzle? How did he come up with the 12 and 19, so that the answer worked out so neatly to a diameter of 15 (radius of 7.5)? Based on my playing around with this puzzle, I wouldn’t have expected a nice answer — so that shocked me. I would have anticipated nice side lengths and an ugly diameter, or ugly side lengths and a nice diameter.

Finally: If you like puzzles like this, you might want to google “Sangaku” and look at the twitter feed of Catriona Agg.

[1] At this point, I had a small detour where I briefly tried to work this problem on a coordinate plane, where I was finding the intersection of the two lines to find the location of the center of the circle, point I, based on the coordinates of G… but when I realized that once I had the intersection point, I’d still have to find find the right coordinates for G to make the circle tangent to the edges, I realized that would be annoying. So I abandoned the coordinate plane work, though I could always return to it if I needed.

# Two Problems that Got Me To Think

Here are two problems that have gotten me to think a lot.

The first one came from my Precalculus co-teacher James. We had been finishing up our unit on combinatorics and also creating new groups, and he devised a great question. So here’s the two-part problem I posed to my kids:

First Problem: We have a class of 14 students, with two groups of 3 and two groups of 4. If I were to have a computer program randomly create new groups: (a) what is the total number of different configurations/outcomes we could have? (b) what is the probability that your entire group was the exact same if you were in a 4-person group?

I thought I solved it successfully and was feeling really confident. Then James told me I was wrong. Then I tried but didn’t understand his logic. So I made a simpler case, and then I thought I understood it. My brain hurt so much. I kept switching back and forth between a couple different answers. It was marvelous! Finally, I felt like I understood things and felt confident. I shared it with my class, and lo and behold, a couple students got what I got, and a couple students didn’t. But the students who didn’t convinced me with their logic. And then I shared their thinking with James, who didn’t have the same answer, and he too was convinced. And I thoroughly enjoyed being wrong and telling the kids that this problem messed with my head, and they helped me see the light!

The second problem came from a student who emailed me about wanting to become a better problem solver. And they shared this old entrance exam for this summer camp they were thinking of possibly applying for, and wanted some guidance. The problem that I got nerdsniped by and ended up spending hours working on over Thanksgiving break was as follows:

Second Problem:

This is from the 2019 entrance questions for a summer program. I think I was able to successfully solve (a) and (b). And then I think I solve (c) for n=3 and n=4 (and got an answer for n=5, but haven’t proved it is optimal). And I have no way to even start thinking about (d). But what I thought was lovely is how many different places my brain when went trying to think through this problem. And the neat geometric structure that arises out of the setup. (Even though I wasn’t able to fully exploit this structure in my thinking.)

I hope you enjoy thinking about these!

# Archiving some gems from Twitter (April 2019)

I have seen a lot of great stuff on twitter lately, and I’ve missed a lot too, I’m sure. I wanted to just archive some of the things that I’ve saved so they don’t disappear! I also think it might be a benefit for someone who reads this who isn’t on twitter or missed some of these tweets. But that’s just a side benefit. I’m writing this for me!!!

***

Desmos writes interesting job descriptions when they have openings. When someone pointed that out to them, they mentioned that this article on reducing unconscious bias helped informed how they write their job descriptions. It’s pretty great and I highly recommend it if you’re hiring. I have thought a lot about “fit” in the past few years when doing hiring, but it’s tricky to think about it well. I have come to recognize that someone entering our department needs to be open and willing to collaborate and compromise, but also have sympathetic pedagogical beliefs with what our department values (and can’t compromise on those). One way I have tried to avoid it is thinking about these things:

But also I have found it harder to balance these thoughts, which I admittedly have a lot:

Not quite those things, but similar thoughts that get at my own personal views on the what persona/personality traits make an effective teacher. Which I tend to think mirror my own traits. But that’s only because I have these traits because I think they make an effective teacher. But I have worked with enough amazing teachers to know that amazing teachers come in all personas! Just like amazing students don’t all have to have the same personas. But this type of bias is something I am trying to be super cognizant about when on hiring committees.

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I saved this just because I like the question and wanted to work on it. And I can see all kinds of extensions. A formula for n circles? What about spheres? I’m guessing (without working on this problem yet) that this is a classic “low entry point, high ceiling” type problem.

***

I just really liked this quotation, and I need to think about the ways that students can see themselves in the mathematics they do. It is part of a larger thing I want to do which is “humanize math” — but I’m not very good at making it a core part of what I do in the classroom. Small bits here and there humanize and expand what kids think about math, but I’m not there yet. I want to one year leave the classroom and know that kids have looked in the mirror and saw something. (It kind of reminds me in a super literal way of how Elissa Miller put a mirror in her classroom, and I think on the bottom she wrote “mathematician.”)

***

Okay, I love this so much. If you’ve never seen it before, it a great trick. You have someone pick any number between 1 and 63 secretly. They just point to the cards that number is on. In about three seconds, I can tell you your number.

I actually made a set of these cards where the numbers are more jumbled up, so kids don’t see a pattern to it. I do put the powers of 2 in one of the four corners though to make things easier for me. Oh wait, have I said too much?

If you don’t know this trick, or how or why it works, I’m sure you can google it. But I’m going to recommend the awesome book “Math Girls Talk About Integers” (there are a lot of great “Math Girls” books out there, so make sure you get the Integer one.

Not only is the book awesome (and great for kids to read), but it breaks down this trick so well. *Shivers with joy*

***

I was excited with Karen Uhlenbeck won this year’s Abel Prize, the first woman to win it ever! I had my kids read this article in the NYTimes about it, and write down three notes about the article. We started the next class with a “popcorn sharing” of what people wrote down. (I also said that although I liked the article, it was a bit dense and thought it could have been written more lucidly.) One thing that came up in both classes I did this in was what a “minimal surface” was — so I told kids it is a surface with minimal area.

I then showed my kids this short youtube video:

And explained that bubbles, though not “central” to all higher level mathematics, do come up. And then I gave them a question. I’m too lazy to type it out, but watch the first 1 minute and 45 seconds of this video (https://www.youtube.com/watch?v=dAyDi1aa40E) and you’ll see it. Then we talked about some basic solutions. And THEN I revealed the best answer was the answer shown in the video we all watched together.

Of course @toddf9 (Todd Feitelson) used this as inspiration to create his own bubble thingies:

but he also explained how he made them…

and then he EVEN created an awesome desmos activity on this very problem, which I want to archive here for use later: https://teacher.desmos.com/activitybuilder/custom/5cb50bed4dcd045435210d29

(Oh! And Mike Lawler (@mikeandallie) made a mobius strip bubble!)

***

Dylan Kane wrote a nice blogpost about calling on students (and the “popsicle sticks of destiny” — though he doesn’t call them that). My favorite line is this simple question that isn’t about right or wrong:

• After students attempt a problem in groups, or reflect on an idea and share with partners, I call on students asking, “How did your group approach the problem?” or “What is something useful that you or your partner shared?”

It’s so obvious, but even after so many years of teaching, I forget to ask things like this. Or my curriculum isn’t group problem solving based enough for things like this to make sense asking. Or whatever.

***

Questions are good. I might have a kid read this at the start of the year and then have a short conversation about why we’re reading it.

It will get at the problematic idea of “obvious,” and when and how learning happens and more importantly when and how learning doesn’t happen.

***

In case you didn’t know, Desmos has a list of all their mathematicians they use when they anonymize in Activity Builder.

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I can imagine putting this picture on a geometry test as a bonus question and asking them why it makes math teachers all angsty… Plus it made me chuckle!

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I’m so not here yet. Anyone who knows me as a teacher will probably know I’ll probably never get here. I’m such a stickler for making the use of every second of classtime.

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Crystal Lancour (@lancour28) tweeted out a slide from a session led by Robert Berry (NCTM president) which had this very powerful slide:

Four rights of the learner in the mathematics classroom

1. The right to be confused and to share their confusions with each other and the teacher
2. The right to claim a mistake
3. The right to speak, listen, and be heard
4. The right to write, do, and represent only what makes sense to you

***

Love the idea of using marbles/paint to draw parabolas (click here to go to the original tweet and watch the video — it’s not a static picture).

***

Bree Pickford-Murray (@btwnthenumbers) gave a talk at NCTM about a team-taught math and humanities course called “Math and Democracy.” Not only did she share her slides (like *right after* the talk) but also she links to her entire curriculum in a google folder. SUPERSTAR!!!

I’ve gone to a few talks about math and gerrymandering (both at MoMATH and NYU) and listened to a number of supreme court oral arguments on these cases. It’s fascinating!

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I just finished teaching “shape of a graph” in calculus. But I wish I had developed some activities like this, to make it interactive:

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I’ve literally been preparing to give a talk next month for… months now. And this one stupid tweet summarized the talk. Thanks.

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I have so many more things I can post, but I’m now tired. So this will be the end.

# Double Angle Formulae

I posted this on my Adv. Precalculus google classroom site. I don’t know if I’ll get any responses, but I loved the problem, so I thought I’d share it here.

***

I mentioned in class that I had stumbled across a beautiful different proof for the double angle formulae for sine and cosine, and I would post it to the classroom. But instead of *giving* you the proof, I thought I’d share it as an (optional) challenge. Can you use this diagram to derive the formulae? You are going to have to remember a tiiiiny bit of geometry! I already included one bit (the 2*theta) using the “inscribed angle theorem.”

If you do solve it, please share it with me! If you attempt it but get stuck, feel free to show me and I can nudge you along!

***

Below this fold, I’m posting an image of my solutions! But I say to get maximal enjoyment, you don’t look further, take out a piece of paper, and take a stab at this!

# Problem Solving with Trig

So I’m at #TMC17 and Rachel Kernodle nerdsniped me. Or rather, I asked to be nerdsniped. Her session is at a time when there were a lot of other amazing sessions I wanted to go to, so I wanted to know if hers was one where I could hear about it and get the gist of things instead of attending. After some internal debate, she said that since it involved working on a problem, and then using that problem solving to frame the session, the answer was maaaaybe not. But then she thought: maybe I can try the problem on you and see how it goes. As long as you’re willing to put in the time to problem solve. Of course I said yes.

First, you can see her session description, which then framed how I approached the problem:

And then this is what she gave me (but it was hand drawn):

From the session description, I knew I had to find the ratio of the side lengths, so I could find exact trig values for angles other than 30, 60, 90, 45.

Rachel also gave me a “hint page” which she told me to look at when I was stuck (and to time how long it took me before I opened it). Let’s just say I’m extremely stubborn, and so as long as I think I have the capability to solve something and I am not completely stuck, I knew I wasn’t going to open it. Turns out my stubbornness paid off, and I ended up solving it.

In this post, I wanted to write a little bit about my experience with the problem. Because now when I look at that triangle, I have an duh, there’s an obvious approach to use here and everything I know points at that obvious approach. And the answer feels really obvious too. It is funny that I’m almost embarrassed to post this because there are going to be people who see it right away, and I worry (irrationally) (math pun) that they are going to judge me for not seeing it as quickly as they did. Even though I know being good at math has nothing to do with speed. And that it was important to go through the steps I did!

It took me over an hour to solve this problem. I had to do a lot of play and make a lot of random leaps before I stumbled across the “obvious approach.”  And I needed to do that in order for me to mine it for lots of things. It was true problem solving. And I know I really deeply understand this because at first the problem looked flummoxing and interesting, and now it looks obvious and somewhat trite. That’s my metric of how I know I deeply understand something. There are still certain things that I teach that I don’t deeply understand: like how the cross product of two 3D vectors yields a third vector perpendicular to the original two. I have done the math, but it’s non-obvious to me why the crazy way we compute cross products give us something perpendicular.(When I only understand something by doing brute algebra, I rarely feel like I get it.)

I’m going to try to outline the messiness that was my thought process in this triangle problem, to show/archive the messiness that is problem solving.

1. The first thing I noticed was 36 and 36 sum to 72. So I was like: obviously put two of those figures together, and just play around. Something nice will happen. I remember when seeing the problem that approach felt immediate, obvious, and would lead to the solution. I was like yes! I have an inroad! This is going to rock, and I’m going to solve it quickly! And I’ll even impress Rachel!

That appraoch didn’t work. Nothing popped out. I saw 54s and 18s and 144s pop out. But those weren’t angles that helped me. But I did then realize something nice… 36 is a tenth of 360! So I was going to use a circle somehow in this solution. Obviously!

2. So I drew this:

and I was like, I have something here! But after looking around, I was getting less. You can see I was trying to draw in some other lines lightly and play around — I thought maybe creating other triangles within these triangles would work. But nothing seemed to pop out. At one point, I thought I had possibly created an equilateral triangle in this (even though I saw one of the angles was 72! I was clearly desperate!). I started to get dejected at this point. I knew the circle had something to do with it…
3. But seeing that 54s and 18s and 36s and 72s kept appearing, I thought maybe algebraically I should play around with the numbers (adding in 180 also, since I can draw a straight line wherever) to see if algebraically I could get a 30, 60, or 45. I tried adding and subtracting numbers from the set {18, 36, 54, 72, 180} looking for 30, 60, or 45. I figured if I could somehow do that, then I could find a diagram that would have angles I could get side relationships from. And then like a domino effect, I could get others. I don’t know. But after like 2 seconds, I got bored with this and didn’t see it as very efficient. My intuition was strongly saying I was going in the wrong direction. So I stopped:

4. At this point, I was pretty dejected. I was slightly losing interest in the problem, thinking it was too hard for me. I tried to “force” a 60 degree angle in a diagram of that original blasted triangle. Hope! And then hope dashed!

5. Damnit! I know the circle had something to do with it. It is just too nice to abandon the circle! Maybe…

At first I drew all ten vertices for a 10-gon. I started connecting them in different ways. I thought I could exploit the chord-chord theorem in geometry, but that wasn’t good. I tried in that second diagram to extract part of the circle diagram and investigate it more. And the third was just more of the same. At one point, I was like $e^{i\theta}=\cos\theta+i\sin\theta$ and was thinking I could somehow think of this as a problem on the complex plane, where each vertex was $e^{ni\pi/5}$ and then look at the real parts for the x-coordinate and the imaginary parts for the y-coordinate. Clearly my mind was whirring, and I was going anywhere and everywhere. I actually thought maybe this complex plane thing seems ugly but it will be so elegant. But then I realized I didn’t know where to go if I labeled each of the points on the complex plane. Done and done and doneAt this point I put the problem away. Nothing was working.

6. But after a minute, I couldn’t let it go! I wanted to solve it!!! So I went back. I thought I was getting too complicated, so I went simple.

Nope. Didn’t help. But for some reason, this diagram and looking at the 72 reminded me of something I hadn’t thought of before. This is the leap that helped me get to the answer. And I can’t quite explain why this diagram sparked this leap. Which sucks because this is that moment that led to the rest of the problem for me! But I immediately remembered something about 72s and pentagons. And it hit me.

7. So I drew what this connection was. My brain was whirring, and I was somewhere good…

I remembered the 72 degree angle appeared in a star. And this star was related to a pentagon. And that the pentagon had something about the golden ratio tied up in it. So I knew that maybe the golden ratio was involved in the answer. And where does the golden ratio appear? When there are similar triangles and proportions. I had my new approach and my inroad that I thought would work. Two triangles next to each other failed. Circles failed. But star/pentagon might work!

8. So I looked at the original triangle and tried to figure out where I could find a similar triangle. And so I drew one line and created a similar triangle. I labeled the two legs as having length “1.”

Initially, I was thinking I could do something with the law of sines. Because if you think about it, this is the ASS case — where you have that 36 degrees (circled), the side I labeled 1 (circled), and the other side I labeled y (circled). But you note that last side could be in two different places, which is why there are two ys circled. I still think there is something fun that I could do with this. But as I was doing this, I realized I was making things more complicated.

I knew that the golden ratio came out of a proportion. So I abandoned the law of sines for the proportion. I simply set up a proportion with the two similar triangles. I first found “?” by doing $1/y=y/?$. So ? was $y^2$This was exciting. I knew the golden ratio came out of solving a quadratic. Yeeeeee! At this point, my excitement was growing because I was fairly confident I was almost at the solution.

Then I labeled the part of the leg that wasn’t ? as $1-y^2$ (since the whole leg length was 1). Finally I looked at the third triangle in the diagram that wasn’t similar to the original triangle. It was isosceles and has legs of $y$ and $1-y^2$ so I set them equal and solved and not-quite-the-golden-ratio came out! (There was a mistake I made where I set $y^2=1-y^2$ and got $y=\sqrt{2}/2$. But I then found it and rewrote the equation $y=1-y^2$. This was the most depressing part of it. Because I couldn’t find my error because I was so tired. I went through my work multiple times and nothing. But taking some time away and then looking with fresh eyes, it was like: doh!)

And so that was the end. I found if the original triangle had leg lengths of 1, the base was going to have a length of $\sqrt{5}/2-1/2$.

I was so proud. I was on cloud nine. I was telling everyone! SO COOL!!!

It probably took me in total 90 minutes or so from start to finish. So many false starts at the beginning, and one depressing transcription error that I couldn’t find.

The point of this post isn’t to teach someone the solution to the problem. I could have written something much easier. (See we can draw this auxiliary line to create similar triangles. We use proportions since we have similar triangles. Then exploit the new isosceles triangle by setting the leg lengths equal to each other.) But that’s whitewashing all that went into the problem. It’s like a math paper or a science paper. It is a distillation of so freaking much. It was to capture what it’s like to not know something, and how my brain worked in trying to get to figure something out. To show what’s behind a solution.

# Graham’s Number

TL;DR: If you have an extra 45-60 minute class and want to expose your 9th/10th/11th/12th graders to a mindblowingly huge number and show them a bit about modern mathematics, this might be an option!

In one of my precalculus classes, a few kids wanted to learn about infinity after I mentioned that there were different kinds of infinity. So, like a fool, I promised them that I would try to build a 30 minute or so lesson about infinity into our curriculum.

As I started to try to draft it — the initial idea was to get some pretty concrete thinkers to really understand Cantor’s diagonalization argument — I decided to build up to the idea of infinity by first talking about super crazy large numbers. And that’s where my plan got totally derailed. Stupid brain. At the end of two hours, I had a lesson on a crazy large number, and nothing on infinity. You know, when that “warm up” question takes the whole class? That’s like what happened here… But obvi I was stoked to actually try it out in the classroom.

In this post, I’m going to show you what the lesson was, and how I went through it, with some advice for you in case you want to try it. I could see this working for any level of kid in high school. Now to be clear, to do this right, you probably need more than 30 minutes. In total, I took 35 minutes one day, and 20 minutes the next day. Was it worth it? Since one of my goals as a math teacher is to try to build in gaspable moments and have kids expand their understand of what math is (outside of a traditional high school curriculum): yes. Yes, yes, yes. Kids were engaged, there were a few mouths slightly agape at times. Now is it one of my favorite things I’ve created and am I going to use it every year because I can’t imagine not doing it? Nah.

We started with a prompt I stole from @calcdave ages ago when doing limits in calculus.

Kids started writing lots of 9s. Some started using multiplication. Others exponentiation. Quite a few of them, strangely, used scientific notation. But I suppose that made sense because that’s when they’d seen large numbers, like Avagadros number! I told them they could use any mathematical operations they wanted. After a few minutes, I also kinda mentioned that they know a pretty powerful math operation from the start of the school year (when we did combinatorics). So a few kids threw in some factorial symbols. Then I had kids share strategies.

Then I returned to the idea of factorials and asked kids to remind me what $5!$ was. Then I wrote $5!!$. And we talked about what that meant ($120!$). And then $5!!!$ etc. FYI: this idea of repeating an operation is important as we move on, so I wouldn’t skip it! They’ll see it again in when they watch the video (see below). While doing this, I had kids enter $5!$ on their calculator. And then try to enter $120!$. Their calculators give an error.

Yup, that number is super big.

Then I introduced the goal for the lesson: to understand a super huge number. Not just any super huge number, but a particular one that is crazy big — but actually was used in a real mathematical proof. And to understand what was being proved.

Lights go off, and we watch the following video on Graham’s number. Actually, wait, before starting I mention that I don’t totally follow everything in the video, and it’s okay if they don’t also… The real goal is to understand the enormity of Graham’s number!

I do not show the beginning part of the video (the first 15) because that’s the point of the lesson that happens after the video. While watching this, kids start feeling like “okay, it’s pretty big” and by the end, they’re like “WHOOOOOOAH!”

Now time for the lesson… My aim? To have kids understand what problem Ronald Graham was trying to understand when he came up with his huge number. What’s awesome is that this is a problem my precalculus kids could really grok. But I think geometry kids onwards could get the ideas! (On the way, we learned a bit about graph theory, higher dimensional cubes, and even got to remember a bit about combinations! But that combinations part is optional!)

I handed out colored pencils (each student needed two different colors… ideally blue and red, but it doesn’t really matter). And I set them loose on this question below.

It’s pretty easy to get, so we share a few different answers publicly when kids have had time to try it out. The pressure point for this problem is actually reading that statement and figure out what they’re being asked to do. When working in groups, they almost always get it through talking with each other!

One caveat… While doing this, kids might be confused whether the following diagram “works” or if the blue triangle I noted counts as a real triangle or not:

It doesn’t count as a real triangle since the three vertices of the triangle aren’t three of the original four points given. During class I actually made it a point to find a kid who had this diagram and use that diagram to have a whole class conversation about what counts as a “red triangle” or “blue triangle.”. Making sure kids understand what they’re doing with this question will make the next question go more smoothy!

Now… what we are about to do is super fun. I have kids work on the extension question. They understand the task (because of the previous one). They go to work. I mention it is slightly more challenging.

As they work, kids will raise their hand and ask, with trepidation, if they “got it.” I first look to make sure they connected all the points with lines. (If they didn’t, I explain that every pair of points needs to be connected with a colored line.) Then I look carefully for a red or blue triangle. Sometimes I get visibly super excited as I look, saying “I think you may have gotten it! I think you may… oh… sad!” and then I dash their hopes by pointing out the red or blue triangle I found. (So here’s the kicker: it’s impossible to draw all the line segments without creating a red or blue triangle… so I know in advance that kids are not going to get it… but they don’t know this.) After I find one (or sometimes two!) red or blue triangles, I say “maybe you want to start over, or maybe you want to start modifying your diagram to get rid of the red/blue triangle!” Then they continue working and I go to other students.

(It’s actually nice when students try to modify their drawings, because they see that each time they try to fix one thing, another problem pops up. They being to *see* that something is amiss!)

This takes 7-8 minutes. And you really have to let it play out. You have to ham it up. You have to pretend that there is a solution, and kids are inching towards it. You have to run from kid to kid, when they think they have a solution. It felt in both classes like a mini-contest.

Then, after I see things start to lag, I stop ’em. And then I say: “this is how you can win money from your parents. Because doing this task is impossible [cue groans… let ’em subside…] So you can bet ’em a dollar and say that they can have up to 10 minutes.!That it takes great ingenuity to be successful! What they don’t know is… you’re going to get that dollar! Now we aren’t going to prove that they will always fail, but it has been proven. When you have six or more dots, and you’re coloring all lines between them with one of two colors, you are FORCED to get a red or blue triangle.” [1]

Now we go up a dimension and change things slightly. Again, this is a tough thing to read and understand so I have kids read the new problem aloud. And then say we are going to parse individual parts of it to help us understand it.

And then… class was over. I think at this point we had spent 35 minutes all together. So that night I asked kids to draw all the line segments in the cube, and then answer the following few questions:

These questions help kids understand what the new problem is saying. In essence, we’re looking to see if we can color the lines connecting the eight points of a cube so that we don’t get any “red Xs” or “blue Xs” for “any four points in a plane.” Just like we were avoiding forming “red triangles” and “blue triangles” before when drawing our lines, we’re now trying to avoid forming “red Xs” and “blue Xs”:

So the next day, we go over these questions, and I ask how this new question we’re working on is similar to and different from the old question we were working with. (We also talk about how we can use combinatorics to decide the number of line segments we’d be paining! Like for the cube, it was $_8C_2$ and for the six points it was $_6C_2$ etc. But this was just a neat connection.) And then I said that unlike the previous day where they were asked to do the drawings, I was going to not subject them to the complicated torture of painting all these 28 lines! (I made a quick geogebra applet to show all these lines!) Instead I was going to show them some examples:

It’s funny, but it took kids a long while to find the “red X” in the left hand image. Almost each class had students first point out four points that didn’t form a red X, but was close. But more important was the right hand figure. No matter how hard you look, you will not find a red X or blue X. Conclusion: we can paint these line segments to avoid creating a red X or blue X. Similar to before, when we had four points, we could paint the line segments to avoid having a red triangle or blue triangle!

So now we’re ready to understand the problem Graham was working on. So I introduce the idea of higher dimensional cubes — created by “dragging and connection.” I don’t take forever with this, but kids generally accept it, with a bit of heeing and hawing. More than not believing that it’s possible, kids seem more enthralled about the process of creating higher dimensional cubes by dragging!

And then… like that… we can tie it all together with a little reading:

And… that’s the end! At this point, kids have been exposed to an incomprehensibly large number. And kids have learned a bit more about the context in which this number arose. Now some kid might want to know why we care about higher dimensional cubes with connecting lines painted red/blue. Legit. I did give a bit of a brush off answer, talking about how we all have cell phones, and they are all connected, so if we drew it, we’d have a complex network. And analyzing complex networks is a whole branch of math (graph theory). But that’s pretty much all I had!

In case it’s helpful: the document/handout I used: 2017-04-04 Super Large Numbers (Long Block).

[1] I like framing this in terms of tricking their parents. We’ve been doing that a bunch this year. And although I understand some teachers’ hesitation about lying to their students about math, I think if you frame things well, don’t do it all the time, it can be fine. I don’t think any student felt like I was playing a joke on them or that they couldn’t trust me as their math teacher because of it.

# Waiters, Waiters, everywhere…

Today I was nerdsniped in the math office. My department head and two colleagues were working on this problem. I don’t know where it came from. But golly, did I enjoy it!

Imagine you have a row of waiters all facing forward. Each waiter has a beautiful silver platter that they are carrying. They have to choose: will they hold it directly in front of them, or on their left side, or on their right side? Here’s a diagram showing the three options (I imagine I’m looking down on the waiter.)

Okay, so there is one constraint. Remember the waiters are all standing in a row. So you can’t have the platters crash into each other. So here’s an example of an OK way the waiters could hold their platters, and then a NOT OK way the waiters could hold their platters.

So here’s the question… If you have n waiters standing in a row, how many different ways could they hold their platters?

I am not going to post the answer here, because I like to nerdsnipe! But if you want to check your answer, for 20 waiters, I calculate 267,914,296 different positions!

I bet you will have a lot of fun with this problem. One person in our office came up with many pages of work, and had a very complex approach which yielded some deep insights. She was super psyched about the intricate superstructure she was building. Another person got to review solving a particular type of thingie using matrices (I want to keep things vague so I’m going to use the word thingie to avoid giving anything away). I and another person had the same approach that led to a quick and elegant solution, but left me with rich conceptual questions to pursue. And as I started doing that, I realized that I had accidentally stumbled on the complex approach that the first person had taken.