Modular arithmetic, Khan Academy, and a NY Times puzzle

I’ve had a busy week at work and haven’t been able to put as much time in with the boys as I normally do. Yesterday I spent most of the time with my younger son catching up on some of the problems in our Introduction to Number theory book and ended up with nothing obvious for him to work on for homework. I ended up telling him just to play around on Khan Academy.

This morning I got a bit of a surprise when he asked me what modular arithmetic has to do with cryptography. I was sort of caught off guard by the question – we are on the modular arithmetic section in our number theory book, but I don’t remember the book mentioning anything about cryptography, and I certainly haven’t.

His reason behind the question was interesting – he saw the connection on Khan Academy. Apparently some of the feedback he’d gotten on the modular arithmetic section was that he had made 5% progress towards a cryptography badge (or some other achievement, I don’t really know). It is always surprising what catches a kid’s attention.

This question about applications of modular arithmetic made me switch up what I was planning on talking about with him today. I saw this puzzle in the NY Times posted by Stephen Strogatz yesterday:

If you don’t have time to click the link, here’s the puzzle:

“A chess master is preparing for a tournament. She plays at least one game a day, but no more than 12 games over any seven-day stretch. Can you show that, if she keeps practicing like this for a long time, there will be a series of consecutive days in which she plays exactly 20 games?”

This is a modular arithmetic question in disguise, so I figured that he might like working through this question even though it is certainly not something that I would expect him to be able to solve on his own. I hoped that the surprise connection to modular arithmetic would be interesting to him.

Before diving into the videos, I need to say that this is tough question and I did no prep work at all for this project. I mean none at all – he asked me the question about cryptography as we were walking into the study. Partially as a result of the lack of prep, the conversation is far from polished and it takes us a while to get to the punch line. Still, a couple of little smiles at the end showed me that he found it to be a really interesting puzzle. With a little more work, I think there’s a really fun project for kids hiding in this project somewhere.

We started with a quick review of what it means for two numbers to be equal mod m. The idea here was to remind him of the congruence definition in the book – two numbers are equal mod m if their difference is divisible by m.


With our short review out of the way, we started in on the puzzle. The first step was to write down a sequence of games played on each day that meets the conditions of the problem. For a young kid, just writing down this numbers is an interesting problem.


Now that we had a sequence of games per day written down, we added up the total number of games played from the start. Luckily we didn’t hit 20 exactly, though, trust me, that was just luck! The next step was to look and see if there was ever a sequence of days were she played 20 games exactly. Noticing where the 20 games were hiding was a tough question for him, but not totally inaccessible.


Finally the connection to modular arithmetic. He notices the connection mod 20 (after noticing a connection in mod 10 – ha!). The reason that our list would definitely have a repeat mod 20 took a while to understand (in fact, the bulk of the video is about why a repeat happens), but at least I think he understood the explanation.

I’m really happy to see how excited he was at the end of this little project 🙂


So, a surprise connection from a little work on Khan Academy combined with a neat NY Times puzzle leads to a fun day. Watching this again just now I’d like a 2nd shot at this project, but I’m still really glad we did it. It is neat to see the math you are studying show up in a some surprising places!

Patrick Honner’s challenging parallelogram problem

[post writing note – not much editing on this one. Long day today – I’m going to take a bubble bath instead of editing . . . .]

After taking a day away from our Geometry book to work through a really nice set of exercises that Kate Nowak posted on twitter:

Going through a Kate Nowak Exercise

we started a new section in our Geometry book today – parallelograms.

Well, sort of. Some problems from an old AMC 10 distracted us (in a good way) for about 45 min, so we really only spent about 10 minutes talking about parallelograms. For homework I asked my son to work through the example problems in our book, some of which were pretty challenging. For example, one problem asked you to prove that if you had a quadrilateral ABCD with AB = CD and AD = BC, then ABCD must be a parallelogram.

I posted our daily video on twitter today and mentioned that we talked about parallelograms. Patrick Honner responded by mentioning a really nice challenge problem:

The problem itself is this: If a quadrilateral has a pair of opposite, congruent sides and a pair of opposite, congruent angles, is it a parallelogram?

I’d rate this problem as being more challenging than any of the example problems that my son worked on today, but I asked him to take a look at it during the day anyway so that we could talk it through tonight.

Here’s the first part of the talk – about 4 min long. He has a little bit of trouble getting going, but we do get to talk about some interesting properties:

(1) He knows that the statement would be true for a parallelogram,
(2) He gets a little confused about “the same” meaning “parallel”
(3) Next he draws the picture in a configuration where the two remaining sides are perpendicular to one of the other sides. He is able to see that the two sides that are supposed to be equal, couldn’t be equal in this configuration (without being parallel).

Our conversation continued after this, but I broke it into two pieces for ease of watching:


We continued the conversation with a new picture. In this piece

(1) We start with a little confusion about the remaining two angles. He initially believes that they must be the same.
(2) After we clear up that confusion there’s a nice little coincidence – he has very nearly drawn the picture you need to solve the problem.
(3) The lucky picture allows us to take a guess that the shape is not required to be a parallelogram.


We took a break for dinner, and after dinner we returned to the project using our Zometool set. I wasn’t sure how this part was going to go, but it actually went really well. My fear was that the Zometool set would not plug in properly in the configurations that we needed (which is what did happen), so I purposely started with some large side lengths to mask the potential problems created by the pieces not plugging together properly. At first we tried to make the two opposite angles really small. We found that there was a way to make a parallelogram, but it seemed like no other quadrilateral would work with these two small opposite angles.

Oops – reviewing the movie I see that I missed an opportunity to talk about a non-convex quadrilateral. Next time . . . .

The next angle we tried turned out to be 60 degrees just by luck of how the blue struts plug together. Here we were able to find a configuration that seemed to work. Unluckily it did not work in a configuration where all of the blue struts plugged into each other, but it is still clear that the configuration satisfies the conditions of the problem.


Although this problem was a little over my son’s head, I enjoyed talking about it and hearing his thoughts about how to approach it. He’s having a little bit of trouble with the common math mistake of assuming the things you want to be true are already true. Hopefully working through problems like this helps him learn how to avoid that sort of mistake.

I was also really happy to see how much insight our Zometool set provided in this problem. A more typical use of the Zometool set for us has been building elaborate 3D shapes, but here the use was providing us with two side and two angles that we knew were equal. Having those easy-to-make manipulatives allowed us to get to the heart of the geometry really quickly. Fun project!