**2nd and 3rd grade Circle**

This circle met for just 30 minutes. I know that the Collatz conjecture is dependably fun for kids this age, so that was our main activity. I asked the kids what they thought mathematicians do, and got a reasonable answer, but saw that there wouldn't be time for useful discussion. So I said a bit about math being like a game for mathematicians, and how fun it was to come up with a new puzzle.

In 1937 (I just said it was about a hundred years ago), Lothar Collatz came up with this puzzle/game:

- Pick a number.
- If it's even, cut it in half. Write your new number.
- If it's odd, triple it and add one. Write your new number.
- (We drew an arrow from each number to the next.)
- Repeat until you get back to a number you've already written.

Collatz conjectured (guessed) that the sequence would end up at 1, no mater what number you started with, but he couldn't prove his conjecture. Mathematicians have tried to prove this for over 75 years, and it is still an open question. (It is very likely to be true. Using computers, people have tested every number up to and past 5 quintillion.)

As I expected, the kids loved it. At the end, I showed them a "mind reading" trick.

- Pick a number from 1 to 31. Don't say it, just keep it in your brain.
- (I pretend I'm sucking their thoughts over to my own head.)
- Now show me which of these five cards it's on.
- (I barely glance at the cards.)
- Your number is ___.

**4th and 5th grade Circle**

This circle met for an hour and a half. My plan was to analyze Spot It with them. (I've written at least 4 posts on using Spot It for math circles. Search on Spot It to find them.) We started out playing the game for about 15 minutes, which they all enjoyed.

The problem was, half of them had done this last year in their math class at Nueva! Luckily, one girl had come early and I had shown her the number trick. I asked her if she wanted to teach it to the others. She did.

I split the group in two, and she showed her group the number trick, while my group started thinking about the game. I had one boy who answered every question very quickly, and asking him to slow down didn't help. So, after we had figured out that there would be 57 different pictures, I got out the half-size index cards and suggested they make their own decks, with 4 pictures per card. Or, if they weren't into drawing pictures, 4 numbers per card. They worked hard at trying to make a deck where each card matched every other card on exactly one picture. Towards the end, they wanted to play with the number trick too.

About halfway through the girl who led the other group came over and said, "The number trick is done." So I joined their group for a bit, and asked, "Why does it work?" A few parents were there, thinking about it with their kids. I should have asked them to work with all the kids (about 6 of them), but didn't think to say it. A few kids wandered away, to the puzzle room, no doubt.

The kids who stayed worked hard on the problems and had fun. I had a great time.

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