Candy for Everyone

Several students are sitting in a circle. Each student has an even (though not necessarily the same) number of wrapped pieces of candy. On a signal, each student passes half of his or her trove to the student on his or her right. Between signals, the teacher (reaching into an inexhaustible goody bag) gives any student left with an odd number of candies an extra piece to make the number even before repeating the maneuver.

What happens to the distribution of candy among the students if the maneuver is performed over and over again?

Will one person end up with all the candies? Will everyone's trove grow larger and larger as more and more extra candies are added from the reserve? Will the number of candies stabilize and eventually even out among the students? Might an oscillatory pattern occur, with clumps of candy moving around the circle with each iteration? Does what happens depend on the number of people involved in the exchanges or on the initial distribution of candies?

Ron Lancaster, a math teacher in Hamilton, Ontario, first encountered the candy problem in an article by James Tanton of Merrimack College, Massachusetts, in the September, 1999 issue of Math Horizons, under the heading "Iterated Sharing." Lancaster found himself captivated by the problem, and he soon wrote a little program for his TI-83 graphing calculator to do some simulations and see what happens. It also occurred to him that the problem would make a wonderful class activity for middle-school math students, requiring a nice blend of conjecture, experimentation (or simulation), and proof. Lancaster worked with teachers Carly Ziniuk and Sharon Djordjevic of Bishop Strachan School in Toronto to try the idea out with several seventh-grade classes. "The students found the problem to be highly engaging and interesting," Lancaster reports. "We were amazed at the types of questions they asked and answered."

The students were arranged in a circle, and a random number generator determined how many pieces (an even number up to 10) each one started with. "It took almost an entire period with each class to work through the iterations," Lancaster notes. "Along the way, we listened to the students discuss what they thought would eventually happen, and we challenged them to think about certain aspects of the problem." For example, it proved useful to focus on the maximum and minimum number of candies held by individuals after each round.

The students recorded the number of candies that they individually held during each round, the total number of candies handed out so far, and other data on worksheets prepared by Ziniuk. Before the activity was over, they came to the conclusion that everyone would eventually end up with the same number of candies.

That's certainly true when everyone starts with the same (even) number of candies. Nothing happens to those numbers on each subsequent turn. It's then possible to explore what happens for somewhat more varied initial distributions. That's where the bounding effect of maxima and minima comes into play.

You can find the complete mathematical argument in the book Over and Over Again by Gengzhe Chang and Thomas W. Sederberg (see chapter 6), along with other entertaining and intriguing problems involving iteration and transformation.

"There are myriad ways to explore this problem further," Lancaster and Ziniuk remark in a paper presented at a conference for math teachers last fall.

You can ponder the effect of small changes in the rules, for example. What would happen if the sharing pattern is varied? Suppose each person gives half of his or her candy to the person on the right and half to the person on the left. Or what would happen if each person ending up with an odd number of candies eats the extra one (instead of obtaining one from the reserve stock) to return the total to an even number?

And, in the original problem, can you predict from the initial number of people in the circle and the initial distribution of candy, how many pieces each participant gets at the end? How many iterations does it take to reach a stable pattern?

There's much food for mathematical thought here!

References:

Chang, G., and T.W. Sederberg. 1997. Over and Over Again. Washington, D.C.: Mathematical Association of America.

Tanton, J. 1999. A half-dozen mathematical activities to try with friends. Math Horizons 7(September):26.

Ron Lancaster can be reached at ron2718@netaccess.on.ca.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.