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International Journal of Mathematical Education in

Science and Technology, Vol. 42, No. 5, 15 July 2011, 615–623

Riffs on the infinite ping-pong ball conundrum

Ami Mamolo

a

*and Tristram Bogart

b

a

Faculty of Education, Simon Fraser University, 8888 University Drive, Burnaby, BC V5A

1S6, Canada;

b

Department of Mathematics and Statistics, Queen’s University, 99

University Avenue, Kingston, ON K7L 3N6, Canada

(Received 26 September 2010)

This article presents a novel re-conceptualisation to a well-known

problem – The Ping-Pong Ball Conundrum. We introduce a variant of

this super-task by considering it through the lens of ‘measuring infinity’ – a

conceptualisation of infinity that extrapolates measuring properties of

numbers, rather than cardinal properties. This approach is consistent with

a nonstandard analysis approach to infinite numbers, and gives credence to

the intuitive (but otherwise normatively incorrect) resolution. We explore

the mathematical motivation and consequences of this variant, as well as

offer further ‘riffs’ on the infinite ball problem for consideration.

Keywords: infinity; measuring infinity; ping-pong balls; variations

1. Introduction

Imagine the following scenario:

You have an infinite set of numbered ping-pong balls and a very large barrel and you are

about to embark on an experiment,which lasts 60 seconds.In 30 seconds,you place the

first 10 balls into the barrel and remove the ball numbered 1. In half of the remaining time,

you place the next 10 balls into the barrel and remove ball number 2. Again,in half the

remaining time (and working more and more quickly), you place balls numbered 21 to 30 in

the barrel and remove ball number 3and so on.After the experiment is over,at the end of

the 60 seconds,how many ping-pong balls remain in the barrel?

This thought experiment is a task which occurs within a finite interval of time, yet

which involves infinitely many steps – a phenomenon known as a super-task. The

complexity of this super-task, and the challenges that university students face when

addressing it have been explored in current mathematics education research [1].

In this article, we explore some variations on the ping-pong ball conundrum,

focusing on one in particular which presents and resolves the paradox in a

‘nonstandard’ way. We begin with a detailed resolution of the paradox.

*Corresponding author. Email: amamolo@sfu.ca

ISSN 0020–739X print/ISSN 1464–5211 online

ß2011 Taylor & Francis

DOI: 10.1080/0020739X.2011.562317

http://www.informaworld.com

2. The ping-pong ball conundrum resolution

The answer may come as a bit of a surprise: at the end of 60 seconds, zero ping-pong

balls remain in the barrel. Why?

The question ‘how many?’ is one of cardinality. That is, ‘how many balls

remain in the barrel?’ is interpreted as ‘what is the cardinality of the set of balls which

are not removed from the barrel?’ To compare the cardinalities of infinite sets, we

rely on the work of Cantor [2], who defined two sets to have the same cardinality if

and only if they can be put in one-to-one correspondence. Unpacking the

experiment, we see there are three infinite sets to consider: the in-going ping-pong

balls, the out-going ping-pong balls and the intervals of time. To answer the

question ‘how many’ we look to the existence (or not) of correspondences

between pairs of sets.

We use the following notation: let Arepresent the set of balls added to the barrel,

Rthe set of balls removed from the barrel, Bthe set of balls remaining in the barrel

and Tthe set of time intervals. To determine the number of balls which are not

removed from the barrel (the cardinality of B) we begin by comparing the

cardinalities of A,R, and T.

The sets Aand R, numbered as they are, both correspond to the set of natural

numbers. The set of balls removed and the set of time intervals can be represented,

respectively, as

R¼f1, 2, 3, ...g,T¼f1=2, 1=4, 1=8, ...g

(where each element in Tcorresponds to the length of the time interval in minutes).

These two sets can be put into one-to-one correspondence by pairing each n2Rwith

(1/2)

n

2T. This correspondence assures us that a ball is removed at each of the time

intervals.

These facts are necessary but not sufficient to resolve the paradox. An essential

feature of this thought experiment is the ordering of the out-going balls. In order for

the barrel to be empty at the end of the experiment, the ping-pong balls must be

removed consecutively, beginning from ball numbered 1. Consequently, there will be

a specific time at which each in-going ball is removed: ball numbered nis removed at

time (1/2)

n

, for all n2N. Thus, at the end of the experiment the barrel will indeed

be empty.

Now, what if we change the order of the out-going balls?

3. A minor riff

This experiment begins in the same way as the last,with one slight difference:In 30

seconds,the first 10 balls are placed in the barrel and the ball numbered 1is removed.In

half of the remaining time,the next 10 balls are placed in the barrel and the ball numbered

11 is removed.Again,in half the remaining time,balls numbered 21 to 30 are placed in the

barrel,and the ball numbered 21 is removed,and so on.At the end of the 60-second period,

how many ping-pong balls remain in the barrel?

In this variation, the ping-pong balls are not removed in consecutive order.

Instead, the ball numbered 1 is removed during the time interval with length 1/2, ball

11 during the time interval with length 1/4, ball 21 during the time interval with

length 1/8 and so on. There is no time interval in which balls 2 to 10, 12 to 20, 22 to

616 A. Mamolo and T. Bogart

30, and so on, are removed. Thus, despite the one-to-one correspondence between

the natural numbers, the powers of 1/2, and the set {1, 11, 21, ...}, infinitely many

balls remain in the barrel at the end of the 60 seconds. The seemingly minor

distinction between removing balls consecutively versus removing them in a different

order profoundly changes the resolution of the problems: while in one instance zero

balls remain in the barrel, in the other infinitely many remain.

4. Approaches to infinity

There are several challenges associated with comparing cardinalities of infinite sets,

many of which stem from inappropriate generalisations of properties that hold true

for finite sets [3]. The most notable example relates to the ‘part-whole’ line of

reasoning. With finite sets, a natural way to compare cardinalities is with a ‘part-

whole’ argument: if one set is a ‘part’ of another (the ‘whole’) then it is a proper

subset of the latter and will necessarily have a smaller cardinality. With finite sets,

comparisons by ‘part-whole’ arguments are consistent with comparisons by one-to-

one correspondence. However, the same cannot be said for infinite sets. Take as an

example the set of natural numbers and the set of all even natural numbers. The

latter is a proper subset of the former, yet the two can be put into one-to-one

correspondence. This inconsistency was a source of controversy for Galilei [4],

Bolzano [5], and others and was first resolved by Cantor, who introduced rigour to

infinite set comparisons by relying entirely on correspondences to determine relative

cardinalities.

One particular challenge in understanding the ping-pong ball experiment relates

to the different rates of in-going and out-going balls. As discussed in [1], a frequent

response to the (original version of the) problem draws on the fact that in each time

interval, ten balls go into the barrel while only one ball is removed. The associated

intuition is that the number of balls remaining in the barrel must be a multiple of

nine, or ‘91’. This line of reasoning was noted to be both persuasive and coercive.

Despite sophisticated mathematical backgrounds, despite an understanding of the

normative resolution provided in Section 2, and despite a recognition that

‘infinity does not quite behave the same way as finite numbers’, the idea that

more balls should remain in the barrel than were removed persisted for many

undergraduate and graduate students. Mamolo and Zazkis [1] observed that it was

difficult for students to accept that although for every natural number n,9nballs

remain in the barrel at the end of the nth interval, this property fails at the

completion of the experiment.

The persuasiveness of the idea that ‘91’ balls remain in the barrel at the end of

the experiment struck our interest, and we found ourselves wondering whether it

would be possible to reframe the paradox in such a way that this intuitive response

would be correct. Thus arises a new variant of the infinite ball problem, which we

present in the following section.

To give credence to the intuition that more balls added to the barrel than

removed from the barrel would yield a ‘larger infinite’ number of balls at the end of

the experiment, we look, as suggested by Tall [6], to extrapolate measuring properties

of numbers, rather than cardinal properties. Tall observes that certain properties of

infinity which are false from a perspective of cardinal numbers can be understood as

true from a measuring point of view. He explains further that such an interpretation

International Journal of Mathematical Education in Science and Technology 617

may be given a formal foundation by considering nonstandard number systems that

contain infinite and infinitesimal numbers.

Tall considers the case of two line segments, one twice as long as the other. If a

line segment is considered as a set of points, such as a real interval

½a,b¼fx2R:axbg,

then the notion of cardinality does not capture the difference between two segments:

both have the same cardinality. Measure theory provides one solution, but Tall

suggests a less elaborate approach via the notion of an infinitesimal ruler. In the next

few paragraphs we describe this approach, though with a somewhat different

emphasis than Tall’s.

A typical way of measuring length is to use a ruler: if one segment is twice as long

as the other and both are much longer than the distance between ticks on the ruler,

then the difference between the two will be roughly captured. But only roughly: if the

ticks are one centimetre apart and the segments are exactly 26.283 centimetres

and 412.567 centimetres long, what does the measurement tell us? One possibility

is to round off the lengths, estimating them to be 6 and 13 centimetres. But then we

incorrectly perceive the second interval to be more than twice as long as the first.

A better way to use the ruler is to accept the uncertainty: that is, conclude from our

measurements only that the length of the first segment is between b2c¼6 and

d2e¼7 centimetres, and similarly that the length of the second is between b4c¼12

and d4e¼13 centimetres. (the notations bxcand dxe, respectively, denote the largest

integer less than or equal to xand the smallest integer greater than or equal to x).

The amount of uncertainty is not a fundamental feature of the two segments, but

depends on the precision of the ruler. To decrease the uncertainty, we could use a

more sensitive ruler with more ticks, perhaps spaced one millimetre apart. We

would then determine the shorter interval to be between b10(2)c¼62 and

d10(2)e¼63 millimetres long, and the longer to be between b10(4)c¼124

and d10(4)e¼125. The ratio between the two is then known to be at least 124/

63 1.968 and at most 125/62 2.016. The ratio is now bounded in a smaller range

that still contains the correct ratio of exactly two.

Taking this idea to its logical conclusion, consider a ruler with infinitesimally

spaced ticks. That is, suppose the distance between ticks is a ‘number’ that is

greater than zero, but less than any positive real number. Approximating the length

of each interval as before, we obtain by analogy that the first interval is between

b2/cand d2/eand the second between b4/cand d4/eticks long. To estimate

the ratio between the two lengths, note that bxcx1 and dxexþ1 for any

number. So the ratio is at least

4= 1

2= þ1¼4

2þ¼23

2þ423

2ð1Þ

and at most

4= þ1

2= 1¼4þ

2¼2þ3

2

52þ3

2ð2Þ

But if we know the two lengths to be real numbers, then the ratio between them is

also real. Since our lower bound for the ratio is infinitesimally less than 2 and the

618 A. Mamolo and T. Bogart

upper bound infinitesimally greater than 2, we can now conclude that the true ratio,

being real, must be exactly 2.

In this argument, we used quantities that are discrete, though infinitesimal, in

order to solve a continuous problem of length. Section 6 employs a twist on this idea:

we will use continuous notions to solve a problem about quantities that are discrete,

though infinite.

5. A ‘hyper’ riff

Does a number system exist that can give meaning to such calculations as (1) and (2)?

What is needed is an ordered extension field of Rthat contains infinitesimals. If there

is such a field, it must also contain infinite quantities: if is positive but less than

every positive real number, then 1/is greater than every positive real; i.e. it is

infinite. In fact there are many such number systems. Among them are the hyperreal

numbers of Abraham Robinson [7], which we denote by R*. Just as the set of

hyperreals extends R, there is a set of hyperintegers that extends the set Zof integers.

Any positive hyperinteger that is not already in Zis called an infinite integer.

Using these numbers, we propose the following ‘hyper’ riff on the infinite-ball

problem:

Let W be any infinite integer and suppose that you have a collection of balls numbered by

the ‘hyperintegers’ 1,2,3,...,10W.As before,begin by placing balls numbered 1through

10 in a barrel and removing ball number 1in the first thirty seconds.Then place balls

numbered 11 through 20 in the barrel and remove ball number 2in the next fifteen seconds.

Now continue this process up to step W:for every hyperinteger n less than or equal to W,at

time (1/2)

n

from the end of a minute,place balls numbered 10n9, 10n8, ...,10n1,

10n in the barrel and remove the ball numbered n.At the end of the experiment,how many

balls remain in the barrel ?

In this variation, we again have an experiment where infinitely many balls are

added to and removed from a barrel. However, in the context of R* we have a new

way to talk sensibly (and rigorously) about infinity without being restricted to the

notions of cardinality and correspondences. We can talk in terms of measurement,

and in this context, comparisons take on a new ‘look’. This shift has important

consequences that arise in sharp contrast to the first two formulations of the

problem, as we will see in the resolution. However, before resolving this variant of

the ping-pong ball conundrum, we must first develop some key properties of

hyperreal numbers.

Recall that a (positive) hyperreal xis said to be infinitesimal if 0 5x5yfor every

positive real number y,orinfinite if x4yfor every real number y. There are also

elements of R* that are neither infinite nor infinitesimal and yet are not real

numbers. For instance, if 2R* is infinitesimal, then 1 þis such an element,

and so is the quantity 2 ð3=2Þthat we encountered in the context of an

infinitesimal ruler. There are also many different infinite and infinitesimal quantities

in R*. For example, if M2R* is infinite, then so are Mþ1, Mþ500, M3,

2Mand M/2.

Hyperreals were introduced for the purpose of nonstandard analysis, in which

limits are replaced by the use of infinitesimals and infinite sums. In this theory,

derivatives are defined to be quotients of infinitesimals and Riemann integrals to be

infinite sums taken over infinitesimal intervals. Keisler published a calculus textbook

[8] that takes this approach. The main idea of nonstandard analysis is not to

International Journal of Mathematical Education in Science and Technology 619

demonstrate bizarre properties of R* (of which it has many, beginning with its totally

disconnected topology), but to use R* in new and perhaps more intuitive proofs

of properties of R,R

n

, and other standard spaces. For this purpose as well as for our

own, two properties of R* are essential, the extension principle and the transfer

principle.

Informally speaking, the extension principle allows us to generalise the metaphor

of the real number line to the hyperreal line. More specifically, it allows sets,

functions, and relations to be extended from Rto R*. For example, the interval

½a,b¼fx2R:axbg

is extended to the set

fx2R:axbg:

Since aand bare still real numbers, this extended interval does not contain any

infinite quantities, but it does contain the real numbers in the original interval along

with infinitely close neighbours such as aþb

2þfor any infinitesimal epsilon.

The integers Zextend to the hyperintegers Z*, some of which are infinite. The

nonnegative hyperintegers will likewise be denoted by N*. The function f:R!R

given by f(x)¼2xextends to a function f:R*!R*, thus making sense of quantities

such as 2. Since addition and multiplication are functions from RRto R, they

also extend to the hyperreals, making arithmetic possible in the new context.

Extending the order relation on Rmakes R* an ordered field. This is one of many

properties shared by Rand R* but not by more common extensions such as the

complex field Cor the rational function field R(t).

Shifting our attention back to hyperintegers, we can also, for instance, construct

factorials: n!:¼Qn

i¼1ifor any positive hyperinteger n. Then we may go on to

construct binomial coefficients: n

k

¼n!

k!ðnkÞ!for any positive hyperintegers nand k.

The transfer principle guarantees that these extended sets, functions and relations

continue to behave in familiar ways. Specifically, it states that all first-order

properties carry over from Rto R*. A first-order property is one that can be defined

by quantifying only over elements, not over sets. For example, the commutativity

law of addition:

8x,y2R:xþy¼yþx

is first-order, so it applies to R* as well. So do associativity and distributivity, and it

is an easy but important consequence of the transfer principle that R* is in fact a

field. The transfer principle also tells us that the binomial theorem

ðxþ1Þn¼X

n

i¼0

n

k

xk,

once proved for ordinary numbers, also holds for any hyperreal xand

positive hyperinteger n, whether finite or infinite. Properties of Rthat are not

shared by R*, such as connectivity, are not first-order (recall, a set Sis defined to be

connected if there do not exist two nonempty disjoint open subsets of Swhose

union is S).

The calculations at the end of Section 4 implicitly rely on both of these principles.

The extension principle allows us to construct all of the infinite and infinitesimal

620 A. Mamolo and T. Bogart

quantities that are considered, and the transfer principle justifies the arithmetic

in (1) and (2).

6. Solution to the ‘hyper’ riff

We are now in position to solve the nonstandard variation of the infinite-ball

problem. The solution has three basic parts: a description in terms of nonstandard

analysis of the sets of balls, an interpretation of quantity in terms of nonstandard

analysis and an application of that form of quantification to our specific situation.

We first describe the set Aof all balls that are added to the barrel during the

procedure. Recall that at the nth step, we add balls labelled 10n9,

10n8, ...,10n1, 10n. We do this for every hyperinteger nfrom 1 up to W.

Since Wis infinite, this includes one step for every ordinary positive integer, but also

one step for every infinite integer less than or equal to W. As we have seen, there are

many such infinite integers, such as Witself, W1, W2, W10

100

and bW/2c.

To understand the desired set

A¼[

W

n¼1

f10n9, 10n8, ...,10n1, 10ng,

we must use the transfer principle. We begin with the division algorithm on the

ordinary natural numbers. This states that if aand bare natural numbers, then there

exist unique natural numbers qand rsuch that a¼bq þrand 0 rb1. For

example, if a¼56 and b¼12, we would write 56 ¼412 þ8; that is, q¼4 and r¼8.

The division algorithm is a first-order statement about the natural numbers, so by

the transfer principle, exactly the same holds for hypernatural numbers. So to

produce an element of A, we can take b¼10 and any 1 a10W, and conclude that

a¼10qþrfor some unique q2N* and rbetween 0 and 9. Therefore, every

hypernatural number abetween 1 and 10Warises exactly once in the union above,

so, A¼{1, 2, ...,10W1, 10W}.

Next, we must understand the set Rof all balls that are removed from the barrel.

This is easier: for each hypernatural number nfrom 1 to W, we remove ball nat

stage n.SoR¼{1, 2, ...,W1, W}. Thus the set Bof balls that remain at the end of

the procedure is

AnR¼f1, 2, ...,10W1, 10Wgnf1, 2, ...,W1, Wg

¼fWþ1, Wþ2, ...,2W,2Wþ1, ...,3W,...,10W1, 10Wg:

Before we complete the solution by analysing the set A\R, let us pause to observe

that there are similarities between this situation and the ‘minor’ riff presented in

Section 3. In both cases, there is an infinite set of balls that have not been removed

from the barrel. In Section 3, the question ‘how many remain?’ required a

comparison of cardinality via one-to-one correspondence. This yielded the

conclusion that the same ‘number’ of balls remain in the barrel as were removed,

despite the intuition that the former is nine times larger than the latter.

Now that we have moved away from the realm of cardinalities, we will find a

more intuitive conclusion – though one not without its own caveat. Namely, we

might be tempted to compare the sets by observing that 10Wballs were added and W

were removed, and thus 10WW¼9Wballs should remain. While the equation

International Journal of Mathematical Education in Science and Technology 621

10WW¼9W

is certainly true in R*, it is not sufficient to answer our question. The numbers W,

9W, and 10Ware not cardinalities, but merely elements in the field R*,

so determining differences is not a meaningful way to describe how many balls

remain in the barrel.

Instead, we must return to the idea of substituting measuring for counting, as

introduced in Section 4. Specifically, instead of counting the elements in the sets A,R

and AnR, we measure the intervals [0, 10W], [0, W] and [W,10W] that they,

respectively, span. The length of an interval [a,b]Ris ba. By an appropriate

application of the transfer principle [7, Section 5.1], the same holds for intervals in

R*, both finite and infinite. In particular, the length of the interval [W,10W]is9W

and the length of [0, W]isW. Lengths (i.e. measures), unlike cardinalities, are

additive in the usual way. Thus it makes perfect sense to say that the first interval is

nine times as long as the second, even though both ‘lengths’ are infinite hyperreals.

In other words, we can informally say that the measure of removed balls is ‘1’ and

the measure of remaining balls is ‘91’.

In fact, the situation is quite similar to that described by Tall. He divides finite

intervals into infinitesimal pieces, while we divide infinite intervals into finite pieces.

But if we scale everything down by a factor of W, we return to the former situation,

with the infinitesimal 1/Wplaying the role of , and with the variation that one

interval is nine times as long as the other, rather than twice as long. Furthermore, our

substitution of the intervals [0, W] and [W,10W] for the discrete sets R¼{1, 2,

3, ...,W1, W}and B¼{Wþ1, Wþ2, ...,10W1, 10W} is the reverse of Tall’s

process of using a discrete set (the ruler ticks) to measure a continuous interval.

An interesting feature of the hyperreal ball problem is that although we perform

more steps of adding and removing balls than in the original version (one step for

each hyperinteger from 1 to W, rather than just one for each natural number), the

last step ends 2

W

seconds before the end of a minute. So the process takes

(infinitesimally) less time than in the original version, which requires exactly a

minute. This is consistent with the common intuition that the experiment should

‘never really reach 60 seconds’ [1].

7. More riffs

We end with some further variations on the infinite ball problem for you to consider.

(a) Begin by placing balls numbered 1 through 9 in a barrel in the first thirty

seconds. Instead of removing a ball, however, simply renumber ball 1 as ball

10. In the next fifteen seconds, add balls numbered 11 through 19 to the

barrel, and renumber ball 2 as ball 20. Continue in this way. At the end of the

experiment, how many balls remain in the barrel? Can you name which ones?

(b) Consider balls numbered by the natural numbers, and place them in the

barrel by exactly the same procedure as in the first two versions of the

problem. However, remove them from the barrel randomly: e.g. at the end of

the first stage, randomly remove one of the balls 1, 2, ..., 10 that were just

placed in the barrel. At the end of the second stage, randomly remove one of

the nineteen balls in the barrel and so on. What is the probability that ball

1 is removed from the barrel at some stage? What about ball 11? What is the

622 A. Mamolo and T. Bogart

expected value of the number of balls remaining in the barrel at the end of the

experiment?

(c) An easier version of the previous riff is to place only two balls, rather than

ten, in the barrel at each stage. Again randomly remove one ball from the

barrel at the end of each stage. What is the probability that ball 1 is removed

at some point during the experiment? What is the expected number of balls

remaining at the barrel at the end of the experiment?

(d) Start with a countably infinite collection of unnumbered balls. In the first

thirty seconds, place ten balls in a barrel and remove one ball. In the next

fifteen seconds, place ten more balls in the barrel and remove one more.

Continue as usual. At the end of the minute, how many balls are in the

barrel? Is this process well-defined?

References

[1] A. Mamolo and R. Zazkis, Paradoxes as a window to infinity, Res. Math. Educ. 10(2)

(2008), pp. 167–182.

[2] G. Cantor, Contributions to the Founding of the Theory of Tranfinite Numbers, (Translated

by P. Jourdain), Dover Publications, New York, 1955.

[3] E. Fischbein, Tacit models of infinity, Educ. Studies Math. 28 (2001), pp. 309–329.

[4] G. Galilei, Dialogues Concerning Two New Sciences, (Translated by H. Crew and A. de

Salvio), Dover Publications Inc, New York, 1914. Available at http://www.knovel.com.

proxy.lib.sfu.ca/knovel2/Toc.jsp?BookID=449

[5] B. Bolzano, Paradoxes of the Infinite (Translated by F. Prihonsky, Introduction by

D. Steele), Routledge and Kegan Paul Ltd, London, 1950.

[6] D. Tall, The notion of infinite measuring number and its relevance in the intuition of infinity,

Educ. Studies Math. 11 (1980), pp. 271–284.

[7] A. Robinson, Non-Standard Analysis, revised ed., North-Holland Publishing Company,

Amsterdam, 1974.

[8] H.J. Keisler, Elementary Calculus: An Infinitesimal Approach, Prindle, Weber, and Schmidt

(1976). Available at http://www.math.wisc.edu/keisler/calc.html

International Journal of Mathematical Education in Science and Technology 623