The 17-animal inheritance puzzle is a

mathematical puzzle Mathematical puzzles make up an integral part of recreational mathematics. They have specific rules, but they do not usually involve competition between two or more players. Instead, to solve such a puzzle, the solver must find a solution that sati ...

involving unequal but fair allocation of indivisible goods, usually stated in terms of inheritance of a number of large animals (17 camels, 17 horses, 17 elephants, etc.) which must be divided in some stated proportion among a number of beneficiaries. It is a common example of an apportionment problem. Despite often being framed as a puzzle, it is more an

anecdote An anecdote is "a story with a point", such as to communicate an abstract idea about a person, place, or thing through the concrete details of a short narrative or to characterize by delineating a specific quirk or trait. Anecdotes may be real ...

about a curious calculation than a problem with a clear mathematical solution. Beyond recreational mathematics and

mathematics education In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out Scholarly method, scholarly research into the transfer of mathematical know ...

, the story has been repeated as a

parable A parable is a succinct, didactic story, in prose or verse, that illustrates one or more instructive lessons or principles. It differs from a fable in that fables employ animals, plants, inanimate objects, or forces of nature as characters, whe ...

with varied metaphorical meanings. Although an ancient origin for the puzzle has often been claimed, it has not been documented. Instead, a version of the puzzle can be traced back to the works of

Mulla Muhammad Mahdi Naraqi Muhammad Mahdi Naraqi () (1715–1795) was a Twelver Shi'i scholar, theologian and moral philosopher. Al-Naraqi was a brilliant thinker of the late 12th/18th and early 13th/l8th century. His son, Molla Ahmad Naraqi, was also a celebrated Twelver s ...

, an 18th-century Iranian philosopher. It entered the western

recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research-and-application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...

literature in the late 19th century. Several mathematicians have formulated different generalizations of the puzzle to numbers other than 17.

Statement

According to the statement of the puzzle, a man dies leaving 17 camels (or other animals) to his three sons, to be divided in the following proportions: the eldest son should inherit of the man's property, the middle son should inherit , and the youngest son should inherit . How should they divide the camels, noting that only a whole live camel has value?

Solution

As usually stated, to solve the puzzle, the three sons ask for the help of another man, often a priest, judge, or other local official. This man solves the puzzle in the following way: he lends the three sons his own camel, so that there are now 18 camels to be divided. That leaves nine camels for the eldest son, six camels for the middle son, and two camels for the youngest son, in the proportions demanded for the inheritance. These 17 camels leave one camel left over, which the judge takes back as his own. This is possible as the sum of the fractions is less than one: + + = . Some sources point out an additional feature of this solution: each son is satisfied, because he receives more camels than his originally-stated inheritance. The eldest son was originally promised only camels, but receives nine; the middle son was promised , but receives six; and the youngest was promised , but receives two.

History

Similar problems of unequal division go back to ancient times, but without the twist of the loan and return of the extra camel. For instance, the

Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057, pBM 10058, and Brooklyn Museum 37.1784Ea-b) is one of the best known examples of ancient Egyptian mathematics. It is one of two well-known mathematical papyri ...

features a problem in which many loaves of bread are to be divided in four different specified proportions. The 17 animals puzzle can be seen as an example of a "completion to unity" problem, of a type found in other examples on this papyrus, in which a set of fractions adding to less than one should be completed, by adding more fractions, to make their total come out to exactly one. Another similar case, involving fractional inheritance in the Roman empire, appears in the writings of

Publius Juventius Celsus Publius Juventius Celsus Titus Aufidius Hoenius Severianus (c. 67 – c. 130) — the son of a little-known jurist of the same name, hence also Celsus filius — was, together with Julian, the most influential ancient Roman jurist of the High Clas ...

, attributed to a case decided by

Salvius Julianus Lucius Octavius Cornelius Publius Salvius Iulianus Aemilianus (c. 110 – c. 170), generally referred to as Salvius Julianus, or Julian the Jurist, or simply Julianus, was a well known and respected jurist, public official, and politician who serv ...

. The problems of fairly subdividing indivisible elements into specified proportions, seen in these inheritance problems, also arise when allocating seats in

electoral system An electoral or voting system is a set of rules used to determine the results of an election. Electoral systems are used in politics to elect governments, while non-political elections may take place in business, nonprofit organizations and inf ...

s based on

proportional representation Proportional representation (PR) refers to any electoral system under which subgroups of an electorate are reflected proportionately in the elected body. The concept applies mainly to political divisions (Political party, political parties) amon ...

. Many similar problems of division into fractions are known from

mathematics in the medieval Islamic world Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important developments o ...

, but "it does not seem that the story of the 17 camels is part of classical Arab-Islamic mathematics". Supposed origins of the problem in the works of

al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...

Fibonacci Leonardo Bonacci ( – ), commonly known as Fibonacci, was an Italians, Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages". The name he is commonly called, ''Fibonacci ...

or Tartaglia also cannot be verified. A "legendary tale" attributes it to 16th-century

Mughal Empire The Mughal Empire was an Early modern period, early modern empire in South Asia. At its peak, the empire stretched from the outer fringes of the Indus River Basin in the west, northern Afghanistan in the northwest, and Kashmir in the north, to ...

minister

Birbal Mahesh Das (; 1528 16 February 1586), popularly known by his title Rajah Birbal (), was an Indian minister and commander of the Mughal Empire. He is mostly known in the Indian subcontinent for the folk tales which focus on his wit. He was app ...

. The earliest documented appearance of the puzzle found by Pierre Ageron, using 17 camels, appears in the work of 18th-century Shiite Iranian philosopher

. By 1850 it had already entered circulation in America, through a travelogue of Mesopotamia published by James Phillips Fletcher. It appeared in ''The Mathematical Monthly'' in 1859, and a version with 17 elephants and a claimed Chinese origin was included in ''Hanky Panky: A Book of Conjuring Tricks'' (London, 1872), edited by William Henry Cremer but often attributed to

Wiljalba Frikell Wiljalba Frikell (June 27, 1818 – October 10, 1903) was a famous magician and author. His stage name was Friedrich Wilhelm Frickel and he also went by W. Frickel and Wiljalba Frickel. He was born in Sagan, Prussia. Thomas Frost wrote about ...

or Henry Llewellyn Williams. The same puzzle subsequently appeared in the late 19th and early 20th centuries in the works of

Henry Dudeney Henry Ernest Dudeney (10 April 1857 – 23 April 1930) was an English author and mathematician who specialised in logic puzzles and mathematical games. He is known as one of the foremost creators of mathematical puzzles. Early life Dudene ...

Sam Loyd Samuel Loyd (January 30, 1841 – April 10, 1911) was an American chess player, chess composer, puzzle author, and recreational mathematics, recreational mathematician. Loyd was born in Philadelphia but raised in New York City. As a chess comp ...

Édouard Lucas __NOTOC__ François Édouard Anatole Lucas (; 4 April 1842 – 3 October 1891) was a French mathematician. Lucas is known for his study of the Fibonacci sequence. The related Lucas sequences and Lucas numbers are named after him. Biography Luc ...

Professor Hoffmann Angelo John Lewis, known pseudonymously as Professor Hoffmann (born 1839–1919), was an English-born barrister, Magic (illusion), illusionist and writer who has been described as "the most prolific and influential magic author and translator unti ...

, and Émile Fourrey, among others. A version with 17 horses circulated as

folklore Folklore is the body of expressive culture shared by a particular group of people, culture or subculture. This includes oral traditions such as Narrative, tales, myths, legends, proverbs, Poetry, poems, jokes, and other oral traditions. This also ...

in mid-20th-century America. A variant of the story has been told with 11 camels, to be divided into , , and . Another variant of the puzzle appears in the book ''

The Man Who Counted ''The Man Who Counted'' (original Portuguese title: ''O Homem que Calculava'') is a book on recreational mathematics and curious word problems by Brazilian writer Júlio César de Mello e Souza, published under the pen name Malba Tahan. Sin ...

'', a mathematical puzzle book originally published in Portuguese by

Júlio César de Mello e Souza Júlio César de Mello e Souza (Rio de Janeiro, May 6, 1895 – Recife, June 18, 1974), was a Brazilian writer and mathematics teacher. He was well known in Brazil and abroad for his books on recreational mathematics, most of them published unde ...

in 1938. This version starts with 35 camels, to be divided in the same proportions as in the 17-camel version. After the hero of the story lends a camel, and the 36 camels are divided among the three brothers, two are left over: one to be returned to the hero, and another given to him as a reward for his cleverness. The endnotes to the English translation of the book cite the 17-camel version of the problem to the works of Fourrey and Gaston Boucheny (1939). Beyond recreational mathematics, the story has been used as the basis for school mathematics lessons, as a

with varied

moral A moral (from Latin ''morālis'') is a message that is conveyed or a lesson to be learned from a story or event. The moral may be left to the hearer, reader, or viewer to determine for themselves, or may be explicitly encapsulated in a maxim. ...

s in religion, law, economics, and politics, and even as a lay-explanation for

catalysis Catalysis () is the increase in rate of a chemical reaction due to an added substance known as a catalyst (). Catalysts are not consumed by the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recycles quick ...

in chemistry.

Generalizations

Paul Stockmeyer, a

computer scientist A computer scientist is a scientist who specializes in the academic study of computer science. Computer scientists typically work on the theoretical side of computation. Although computer scientists can also focus their work and research on ...

, defines a class of similar puzzles for any number

n

of animals, with the property that

n

can be written as a sum of distinct

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

d_1, d_2, \dots

n+1

. In this case, one obtains a puzzle in which the fractions into which the

n

animals should be divided are

\frac, \frac, \dots .

Because the numbers

d_i

have been chosen to divide

n+1

, all of these fractions simplify to

unit fraction A unit fraction is a positive fraction with one as its numerator, 1/. It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When a ...

s. When combined with the judge's share of the animals,

1/(n+1)

, they produce an

Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractions, such as \frac+\frac+\frac. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from eac ...

representation of the number one. The numbers of camels that can be used as the basis for such a puzzle (that is, numbers

n

that can be represented as sums of distinct divisors of

n+1

) form the

integer sequence In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...

S. Naranan, an Indian physicist, seeks a more restricted class of generalized puzzles, with only three terms, and with

n+1

equal to the

least common multiple In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...

of the denominators of the three unit fractions, finding only seven possible triples of fractions that meet these conditions. Brazilian researchers Márcio Luís Ferreira Nascimento and Luiz Barco generalize the problem further, as in the variation with 35 camels, to instances in which more than one camel may be lent and the number returned may be larger than the number lent.

References

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pp. 191–192
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The American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposito ...

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Mathematics Magazine ''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...

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Journal of Mathematics and the Arts The ''Journal of Mathematics and the Arts'' is a quarterly peer-reviewed academic journal that deals with relationship between mathematics and the arts. The journal was established in 2007 and is published by Taylor & Francis. The editor-in-chief ...

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Math Horizons ''Math Horizons'' is a magazine aimed at undergraduates interested in mathematics, published by the Mathematical Association of America. It publishes expository articles about "beautiful mathematics" as well as articles about the culture of mathem ...

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Statement

Solution

History

Generalizations

See also

References