Passing Candy

I was away for the last half year, and stopped updating this website. I'm back now, and our junior math circle has already run a couple of times this year, so I'll try to be better and keeping our activities posted.

Two weeks ago we held our first Math Circle of the year. Again we invited children in kindergarten through 3rd grade -- older students could go to other classes, and requested that the young children bring a parent. We had about 20 children show up, and about 10 parents.

Since it was our first session, we went over some rules of Math Circles:

1. Have fun.
2. Help others have fun.
3. Ask questions.
4. Make mistakes.

(One little girl also suggested "listen to the teacher," but I like to stick with the four basics above and let the parents take care of having their children listen.)

The activity of the day, for our first session, involved candy! It was adapted from an activity due to James Tanton, from his book "Solve This."

The children were sitting at round tables, each table with three to six children. I told the children they would be playing a game with candy. I would give each of them some candy, and their job would be to divide their candy exactly in half, to keep half, and pass the other half to the person sitting to their right.

Before I handed out candy, we talked about what it meant to divide something in half. I asked when it would be hard to divide a stack of candy exactly in half, and the children told me that it would be hard if they had an odd number (although you could cut a piece of candy, or bite it, they suggested). We had a little discussion about even and odd numbers.

If you have an even number of candy pieces, like 2, 4, 6, 8, 10, 12 pieces, then you can divide your candy exactly in half.

If you have an odd number of candy pieces, like 1, 3, 5, 7, 9, 11, then you can't divide it in half without cutting it.

So I added a rule: if you have an odd number of candy pieces during our game, raise your hand and I will bring you one more piece, so you then have an even number.

I then passed out exactly 2 pieces of candy to every child in the room. When they were ready, I asked them to divide their pile in half, and keep half and pass the other half to the person on their right. We did this one time, and I asked what had happened. Had anything interesting happened?

Everyone noticed that they all still had 2 pieces of candy. We talked for a minute then about why, and about what would happen if we kept playing the game forever and ever. They all could see that everyone would always have 2 pieces of candy. Then I gave them a challenge. At your table, take the pieces of candy I have given you, and distribute them in a different way so that you can get the most candy from me that is possible! That is, try to make odd numbers, so you can raise your hand, so I will come around and give you more candy.

The children were pretty excited about this game. Several tables had one child pass one piece of candy to the person sitting next to them, so around the table the numbers of candy were 1, 3, 1, 3. Other tables (with an odd number of children) came up with different strategies. When they were ready, I had them divide their stack in half and pass it along. Those with an odd number raised their hands, and I hurried around the room giving extra candy to the students who needed it.

We played several rounds. After a while, at every single table, every child had exactly four pieces of candy. They realized that no matter how many times they kept playing, once everyone had four pieces, everyone would keep exactly four pieces of candy forever.

At this point, I suggested we change the rules. Rather than raise your hand when you had an odd number of candy, eat a piece of candy instead! Again, I had them start over, trying to distribute the candy so they could eat as much as possible!

We started again and played several rounds. After a little while, every table was in the situation that each child had exactly 2 pieces of candy. They realized they wouldn't be eating any more candy.

Now, at this point, I asked the children if they thought they would always end up with the same amount of candy. They weren't sure, so I suggested they try again, with the game of adding candy rather than eating it, and see if they could get a different result by starting by distributing the candy differently. I wandered the room while they tried.

Playing this game basically used up the rest of the time. None of the children were able to find a way to get more candy. I challenged the tables to see if they could figure out why, but we didn't really have the time to pursue this more carefully. (Maybe I'll bring it back for some other activity -- the answer isn't too tricky I think, even for children this age.)

Although we didn't solve all the problems, at the very least the students left knowing how to split small integers in half, how to tell the difference between even and odd, and with candy stuck in their teeth on an early Saturday morning!

We ended with refreshments. (Not candy.)

For further thought: No matter how the children distribute their candies, they will always reach a stable point where all the children have the same even number of candy. Can you figure out why?