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!
