Tonight I was looking for a quick little project since I’ll be on the road for work tomorrow, and talking about the complexity of a few numbers seemed like it would be a lot of fun.

(oh, and sorry for the poor lighting, we were working on the floor of the study while my wife was watching football . . . )

I started with my younger son – he caught on to the idea of complexity quickly and even formed a nice little conjecture about the complexity of prime numbers. The conjecture turned out to be false, but it helped him understand a new way to compute complexity. He even wondered about how you’d compute the complexity of big numbers.

My older son also found computing the complexity of a number to be interesting. We ended up talking for about 8 minutes, so I split the video into two pieces.

He also caught on to the idea fairly quickly and chose to compute the complexity for 10 first. After that he tried 13 and, similar to my younger son, he thought you’d need 13 1’s because 13 is prime. However, with a little more thinking he found a way with 9 1’s and then eventually 8.

In the second half of the conversation I asked him to come up with a few other things to explore. He thought it would be tough to come up with a general formula. I decided to show him one thing that might be a little surprising – if a > b, the complexity of a isn’t necessarily greater than the complexity of b.

After that he wondered what the smallest possible value of the complexity for a two digit number. He had some ideas that led him to believe the smallest complexity would be 7.

To wrap up, we guessed at the value of the complexity for 100.

So, a great post from Richard Green that isn’t too hard to use with kids. It is so fun to be able to share ideas from math research with kids. This idea from Green was especially fun because it got the boys thinking about numbers in ways that are a little different than usual. Amazing what you can do (and what is unsolved!) with just addition and multiplication with 1’s š