Chains (part 2)

Who: J1

When: before bedtime

what: figuring out the rule for the 60-chain and finding other chains

why: because J1 wanted to discuss the chains exploration

J1 was getting more and more concerned as he looked for his homework folder and didn't find it.  Suddenly, his eyes lit up and he held up a crumpled piece of paper: "Daddy, look what I've got!"

It was our 60 chain from earlier in the week.

I suggested we try some other starting points: 0 2, which he then wrote down until we got back to 0 2.
Next, he asked what would happen if we started with 2 3.  I gave him time to think about it and then pointed to 2 3 in the 60-chain and asked what he thought that meant.

Next, I pointed out that our 0 2 chain had a 2 2, 4 4, 6 6, and 8 8. What about other doubles? He found the 60 chain had 1 1, 3 3, 7 7, and 9 9.  So, what was missing? Eventually, we got to this:

I left him with this challenge: there are two other chains, what are they?

Before he would go to sleep, he wanted to discuss a little whether it was easier to find long chains or short ones.  Eventually, he came to the conclusion that long chains might seem more complicated, but they are easier to find. Can you see why he thought that?