## Tuesday, November 3, 2009

### Tootsie Roll Game (“Iterated Sharing” from Solve This)

A group of student sits in a circle, each with a pile of wrapped candies. (Wrapped candy is used because each piece will be handled by many people before being eaten.) Some people have 20 or more, others none, and the rest some number in between. The distribution is arbitrary except for the fact that everyone has been given an even number of pieces. A reserve supply is set aside.
The group now follows these instructions: Give half of your candy to the person on your left and half to the person on your right (and hence receive a supply of candy from the people on your left and right). Do this simultaneously. Now recount your candy supply. If you now have an odd number of pieces, take an extra piece of candy from the reserve supply. This boosts your pile back up to an even number of pieces and everyone is ready to perform the maneuver again.
What happens to the distribution of candy among the group if this maneuver is performed over and over again? Will people be forever taking extra pieces from the center, so everyone’s amount of candy will grow without bound? Or will the distribution stabilize or equalize in some sense? Will one person end up with all the candy? Might “clumps” of candy move around the circle with each iteration or some strange oscillatory pattern emerge? Is it possible to predict what the result will be?
Taking it further: What happens if instead of adding pieces, you eat any odd piece of candy to bring your pile back down to an even number?
Hint: Based on the investigations my class performed, and the patterns they found, I encourage you to examine what the range of distributions is. For example, if you started with 0 – 18 pieces (a range of 18), what happens? And is it different if you started with 0-20 pieces (a range of 20)?
My classes enjoyed this activity, though my advanced class looked into in much further than my lower level class. They tried several rounds to test and prove conjectures. It was a difficult game to play with a class larger than 12 students. I tried putting them into two groups, but this was difficult since I was facilitating both groups. It was also difficult to play with desks in a circle. Playing on the floor or at a large table would have made things easier.

1. This was a wonderful activity, the kind that you could throw at anyone. I used it during advising week:-)

My own investigations focussed on a circle of three people, and I looked at the case where one person had more than the others. I wanted to know how the final shared amount was related to the starting amounts in this case. No results yet, except to say that (1) the longest run so far is 11 rounds; (2) the shortest run is 2 rounds; and (3) two adjacent evens have the same end game, e.g. whether the 3rd person has 22 or 24 extra, or 46 or 48 extra, the endgame is all having 10 extra or 18 extra, respectively.

Ariane also noted that she got the problem from James Tanton's book "Solve This: Math activities for students and clubs." This is the second problem in the book, and a solution to the initial challenge is in the back. They don't attempt to describe the pattern of how the game ends. Instead, they argue that in each round the maximum # anyone holds doesn't exceed the max of the first round. Those holding fewer candies experience growth in their share unless surrounded by people with the same numbers. Since we can't have unsustained growth of integers in a bounded interval, we must admit that all eventually have the same amount.
is bounded above and that the lower bound is always rising.

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3. I've been returning to this problem in conversations over and over. I have a student who promised to make a C++ interactive game of it ... and I'll let you all know if that happens.

I also wanted to note that the problem as stated in Tanton's book says that we only pass HALF our candies to the left, keeping half, and playing with the same rule that if anyone ends with an odd number of candies, one draws an extra from the (inexhaustible) pile of extras.

A little extra play has revealed that in a circle of 10 people where the holding of one player is increased by 2 and the end-state is investigated, there seems to be a critical value beyond with the play --- using Ariane's rules --- becomes oscillatory. At lower levels the limiting value obtained is related to the numbers involved in the oscillating set of values.

4. Yes, this is an awesome activity to engage kids but as said in the article it proven to be hard with my class of 16 students. But I used the given idea in the article to split my class in two groups. Both groups made their own circle and perform the activity.

It turned to be an awesome, fun and educating activity for kids.