An Evil and Odious exercise from Tanya Khovanova

Wandering around the internet this week, I somehow landed on this old blog post from Tanya Khovanova:

Thue-Morse Odiousness

I’d not heard of evil and odious numbers before but Khovanova’s blog post gave me some ideas about how to talk about them with kids. It seemed like a great topic, actually – kids would get to review a little bit about binary and also have an opportunity to make lots of observations / conjectures about patterns in the sequences that came up.

We’ve used a few ideas from Khovanova’s (incredible!) blog before. For example:

Using divisibility rules to build number sense

PISA and V. I. Arnold Questions from Twitter

Also, since evil and odious numbers were invented by John Conway, here are a few of our preview John Conway-inspired projects:

Sharing the Surreal numbers with kids

The various projects in the last link helped me come up with a fun Family Math night for the 4th and 5th graders at my younger son’s school. Watching the kids play around with the game checker stacks and learn about the surreal numbers was one of the most enjoyable moments in math that I’ve ever had.

The Collatz Conjecture and John Conway’s “Amusical” Variation

Playing John Conway’s Game of Life with Kids

So, with that as background, here’s today’s project. First up was introducing the idea of “evil” and “odious” numbers and finding the first couple of numbers in each set:

Next (off camera) we looked at the first 20 positive integers and then on camera we talked about some of the patterns we saw:

Next we studied the process that Khovanova described in her blog. Oversimplifying a little, that process is:

(1) Start with any sequence of positive intergers,
(2) Turn that into a sequence of 0’s and 1’s based on whether or not each integer is odious or evil,
(3) Convert the sequence of 0’s and 1’s back into a sequence of integers which is essentially the increasing sequence of integers with the smallest numbers which share the same odious / evil pattern

There were lots of fun ideas to discover here:

Finally, we went to the computer to investigate a few other sequences. Some short (and a little clumsy . . . sorry) Mathematica code allowed us to look at these sequences pretty quickly:

So, even though this project was a little longer than usual, this was really fun. There are some great opportunities for kids to form and check some conjectures in a setting where the patterns are almost surely different from anything that they’ve ever seen before.

I’m really happy to have stumbled on Khovanova’s great blog post this week!