# 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

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!