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 đź™‚

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!

## Comments

If you do revisit, here are some possible extensions:

Are the numbers in the problem (the “12 games per week” limit and “20 games” target) special? In other words, for any n, can you guarantee a run of games of length n eventually? Or, if you allow the chessmaster to play m games per week, can you still guarantee a run of 20 games?

Perhaps it is just me, but I particularly liked showing that a run of 7 games must occur with the 12 game limit.

This was a fun puzzle. Honestly, I’m not sure I could have solved it without knowing that you found a connection to modular arithmetic. My practice, when I see things like this on your blog is to work through them myself before reading spoilers or watching the videos.