Of Camels, Inheritance, and Unit Fractions

Of Camels, Inheritance,
and Unit Fractions
A
Paul K. Stockmeyer
lawyer friend, who is also a
mathematics enthusiast, recently
related to me this story that had
been a part of a sermon at his
church.
Under the laws of the country in question,
inheritance was as follows. The eldest son took
1/2, the second son took 1/3, and the third son took
1/9. A man died, survived by three sons, and owning 17 camels. The sons began to bicker, because
the law, if applied to the camels, would not be
good for the camels, or the sons. Finally, they went
to the village priest for advice. The priest listened
to the story, and said, “Look, I have a camel. Take
it. Then you’ll be able to work things out.” So
the brothers took the priest’s camel, and divided
up the camels. The eldest son took 1/2 of the 18
camels, or 9 camels. The second son took 1/3, or
6. The third son took 1/9, or 2. And much to their
surprise, they had a camel left over. So they took it
and gave it to the priest.
Readers may be familiar with this story. According
to recreational mathematics historian David
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Singmaster in his unpublished work Sources in
Recreational Mathematics, problems of proportional
division date at least as far back as the Rhind Papyrus
(around 1650 BC), where an example is given of how
to divide 700 loaves in proportion 2/3 : 1/2 : 1/3 : 1/4.
However, the mathematical recreation involving the
loan or donation of an extra beast to avoid the subdivision of individual animals dates only from the late
19th century.
The first appearance of this type of puzzle that
Singmaster records is in the book Hanky Panky: A
Book of Conjuring Tricks, edited by William Henry
Cremer and published in London in 1872. No author is
listed, but the book is often attributed to the German
magician Wiljalba Frikell. Others claim it is by Henry
Llewellyn Williams. The book describes “a Chinese
puzzle” of dividing 17 elephants into parts 1/2, 1/3,
and 1/9.
The puzzle appeared in several other books and
magazine articles before the end of the 19th century,
including those of the great American puzzle master
Sam Loyd and the British puzzle king Henry Dudeney.
Most writers reproduced the 1/2, 1/3, 1/9 version,
with 17 animals of some sort, but a few other versions
also appeared. Most include an imaginative setting in
ancient Arabia or Asia.
Preliminary Explorations
As soon as the church service was over, my lawyer
friend met with the church organist, another mathematics enthusiast, to discuss variations. Can the number of camels be, say, 19? or 34? or 35? Are at least
three heirs required? What about four? Do we need to
change the fractions? They soon contacted me for help.
Can something like this be done with 19 camels?
Yes. If the inheritance shares are 1/2, 1/4, and 1/5
(summing to 19/20) and the estate is 19 camels, then
the donation of one more camel yields an augmented
estate of 20 camels, from which the sons receive 10, 5,
and 4 camels, respectively, leaving one camel to return
to the donor.
Is there a similar scenario with 34 camels? In the
next section, we illustrate a method for showing that
the answer is no.
For an estate of 35 camels, readers can confirm that
estate shares of 1/2, 1/3, 1/12, and 1/18 will divide
up properly after borrowing a 36th camel. Four other
schemes are also possible with 35 camels.
But what exactly is it that we seek? An important
part of mathematics is constructing appropriate definitions, the rules by which we play the game. We start
this process by observing properties that all these
scenarios have in common:
• There is always exactly one borrowed camel
added to the estate to form an augmented estate. The distribution of the augmented estate is
completed with the return of the borrowed camel
to the lender.
• The estate shares sum to less than 1.
• Estate share fractions are applied to the augmented estate, not to the original estate.
• The estate shares are all distinct, with older sons
inheriting more than younger sons.
• The estate shares are fractions with 1 as the numerator. (Such fractions are called unit fractions,
or Egyptian fractions. The ancient Egyptians are
reported to have written all fractions between
zero and one as a sum of distinct unit fractions.)
• No camels are butchered in the distribution
of estate shares. Each share of the augmented
estate is an integral number of camels.
Getting Formal
Suppose we want to find an inheritance situation
with
sons and an estate of
camels. To do so, we
need to find a representation of 1 as a sum of
unit fractions,
fractions for the estate shares of the
sons and one for the donor. The augmented estate will
consist of
camels, and the donor’s share is to be
one camel, so the final fraction must be
To
avoid cutting up camels, all shares must be multiples
of the one-camel share of
In other words,
all shares must be able to be written as unit fractions
where the denominator is a divisor of
Finally,
we want essentially distinct unit fractions. We will
allow one exception: the last son’s share can be one
camel, so the last two fractions can both be
All other fractions should be distinct.
The decisions made in the preceding paragraph are
somewhat arbitrary, and someone else might frame
things differently. Nonetheless, these definitions and
concepts are important, and we repeat them with
proper formality.
Definition: A distinct-share inheritance situation
with
sons and
camels consists of a representation of 1 as the sum of
unit fractions, listed
in decreasing order of fraction size, with the following properties.
Illustration by Gregory Nemec
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1
2
1
3
1
9
1
18
1
1. The last (smallest) fraction must be
2. The denominators of all the fractions must be
divisors of
3. The fractions must be distinct, except that the
last two fractions can both be
The simplest situation is
This corresponds to an inheritance with
son and an estate
of
camel. The augmented estate is two camels,
and the son and the donor each receives half, or one
camel.
There are two cases with two sons. The first representation is
with an estate of five
camels. The augmented estate is six camels, so the
first son gets three, the second son gets two, and the
donor gets back one. The other case with two sons is
where an estate of three camels is
augmented to four. Here the first son gets two camels,
while the second son and the donor each gets one.
The table at right lists the seven situations for three
sons. (Remember, the fourth fraction is for the donor.)
The third row shows the original situation.
There are 52 possible situations with four sons. They
range from
with 1,805
camels down to
with
15 camels. The preparation of full and accurate listings
for more than four sons would certainly be aided by the
use of a computer. Those with programming skills might
1
2
1
3
1
12
1
18
1
36
Estate Shares
1
Camels in
Estate
41
23
17
11
19
11
7
enjoy designing an algorithm to produce such listings.
Rather than organizing situations by the number of
sons, we could try organizing by the number of camels
in the estate. We have seen situations with 1, 3, 5, 7,
11, 15, 17, 19, 23, . . . camels so far. What about 9
camels? 13 camels? 21 camels? Are there any situations with these numbers?
The answer is no. Suppose we had an estate of 9
camels. Then the augmented estate would contain 10
camels, and the last term in the summation would
be 1/10. The denominators of the other terms of our
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summation must all divide 10. The most we can get
is
which is short of 1. With 13
camels, the only allowable terms in the summation are
which again is short of 1. The
case of 21 camels is similar. With C camels, we need
to have lots of divisors so that there are lots of
fractions to choose from in our summation to 1. With
9, 13, or 21 camels, we see that the numbers 10, 14,
and 22 each has only three usable divisors, and that’s
not enough to get fractions adding to 1.
The numbers 10, 14, and 22 are all examples of
so-called deficient numbers—a positive integer n is
deficient if the sum of its divisors (excluding n itself)
then
is less than n. If the divisor sum is exactly
n is said to be almost perfect. If the divisor sum is
exactly n, then n is said to be perfect. Knowledge of
perfect, almost perfect, and deficient numbers is helpful in this investigation.
Questions to Explore
Our results suggest many avenues for further discovery. Here are a few relatively easy questions:
1. Are there any interesting infinite families of
distinct-share inheritance situations? In particular, is there such a situation for every positive
number
of sons?
2. Are there any distinct-share inheritance situations in which the eldest son’s share is 1/3,
rather than 1/2? What about 1/4, or 1/5?
3. Are there any situations where the estate consists of an even number of camels?
4. Is there always a situation with
for
any
Anyone who plays with these questions for very long
will quite likely discover additional interesting questions to explore.
There are other variations that I find interesting.
The first appears in the book Puzzles for Puzzlers by
Jonathan Always, published by Tandem in London in
1971. In this version, the estate is 13 camels, and the
sons are to receive shares of 1/2, 1/3, and 1/4. Note
that here the shares add up to more than 1. The lawyer handling the estate first takes one camel as his fee,
leaving a diminished estate of 12 camels to be distributed. He then dictates that the first son should receive
camels, the second son should receive
camels, and the third son should receive
camels. But this sums to 13 camels, and
the diminished estate has only 12. The lawyer is forced
to return the 13th camel, and everyone else lives happily
ever after.
Finally, there is a variation in the book O Homem
One of my
mentors claimed that research
is what we engage in when we
don’t know the solution to a
problem. Inheritance questions
are wonderful areas where
almost anyone
can enjoy doing research.
que Calculava by the Brazilian writer Júlio César de
Mello e Souza, writing under the pen name Malba
Tahan. An English translation called The Man Who
Counted was published by Norton, New York, in 1993.
This book is a sort of recreational mathematics version of the Arabian Nights. In this variation there are
35 camels, with the usual shares of 1/2, 1/3, and 1/9,
but with both a lawyer and a priest! The lawyer gets
the priest to donate a 36th camel, and then declares
that the sons should receive
camels,
camels, and
camels. This
sums to 34 camels, leaving one to return to the priest
and one for the lawyer to keep as his fee.
Readers are encouraged to explore one or both of
these variations. Start by gathering data—find other
scenarios similar to one of these. List what these scenarios have in common, then create a formal definition
of an inheritance structure of this type. Continue to
explore this new structure as we have done here with
the original. Are there infinite families of numbers for
which you can construct valid situations? Can you rule
out infinite families of numbers?
One of my mentors claimed that research is what we
engage in when we don’t know the solution to a problem. Inheritance questions of the type discussed here
are wonderful areas where almost anyone can enjoy
doing research! n
Paul Stockmeyer is a professor emeritus of computer
science at the College of William and Mary. His retirement activities include serving as an associate editor of
Mathematics Magazine, teaching mathematics-related
courses for senior citizens, and puttering around in
recreational mathematics.
Email: [email protected]
http://dx.doi.org/10.4169/mathhorizons.21.1.8
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