17-animal inheritance puzzle

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Camels in Merzouga Desert.jpg
17 indivisible camels

The 17-animal inheritance puzzle is a mathematical puzzle 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.

Contents

Despite often being framed as a puzzle, it is more an anecdote about a curious calculation than a problem with a clear mathematical solution. [1] Beyond recreational mathematics and mathematics education, the story has been repeated as a parable 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, an 18th-century Iranian philosopher. It entered the western recreational mathematics 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 1⁄2 of the man's property, the middle son should inherit 1⁄3, and the youngest son should inherit 1⁄9. How should they divide the camels, noting that only a whole live camel has value? [2]

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. [2]

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 8+1⁄2 camels, but receives nine; the middle son was promised 5+2⁄3, but receives six; and the youngest was promised 1+8⁄9, but receives two. [3]

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 features a problem in which many loaves of bread are to be divided in four different specified proportions. [2] [4] 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. [5] Another similar case, involving fractional inheritance in the Roman empire, appears in the writings of Publius Juventius Celsus, attributed to a case decided by Salvius Julianus. [6] [7] The problems of fairly subdividing indivisible elements into specified proportions, seen in these inheritance problems, also arise when allocating seats in electoral systems based on proportional representation. [8]

Many similar problems of division into fractions are known from mathematics in the medieval Islamic world, [1] [4] [9] but "it does not seem that the story of the 17 camels is part of classical Arab-Islamic mathematics". [9] Supposed origins of the problem in the works of al-Khwarizmi, Fibonacci or Tartaglia can also not be verified. [10] It has also been attributed to 16th-century Mughal Empire minister Birbal, but only as a "legendary tale". [11] The earliest documented appearance of the puzzle found by Pierre Ageron, using 17 camels, appears in the work of 18th-century Shiite Iranian philosopher Mulla Muhammad Mahdi Naraqi. [9] By 1850 it had already entered circulation in America, through a travelogue of Mesopotamia published by James Phillips Fletcher. [12] [13] It appeared in The Mathematical Monthly in 1859, [10] [14] 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  [ de ] or Henry Llewellyn Williams. [2] [10] The same puzzle subsequently appeared in the late 19th and early 20th centuries in the works of Henry Dudeney, Sam Loyd, [2] Édouard Lucas, [9] Professor Hoffmann, [15] and Émile Fourrey, [16] among others. [17] [18] [19] [20] A version with 17 horses circulated as folklore in mid-20th-century America. [21]

A variant of the story has been told with 11 camels, to be divided into 1⁄2, 1⁄4, and 1⁄6. [22] [23] Another variant of the puzzle appears in the book The Man Who Counted , a mathematical puzzle book originally published in Portuguese by Júlio César de Mello e Souza 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). [10]

Beyond recreational mathematics, the story has been used as the basis for school mathematics lessons, [3] [24] as a parable with varied morals in religion, law, economics, and politics, [19] [25] [26] [27] [28] and even as a lay-explanation for catalysis in chemistry. [29]

Generalizations

Paul Stockmeyer, a computer scientist, defines a class of similar puzzles for any number of animals, with the property that can be written as a sum of distinct divisors of . In this case, one obtains a puzzle in which the fractions into which the animals should be divided are

Because the numbers have been chosen to divide , all of these fractions simplify to unit fractions. When combined with the judge's share of the animals, , they produce an Egyptian fraction representation of the number one. [2]

The numbers of camels that can be used as the basis for such a puzzle (that is, numbers that can be represented as sums of distinct divisors of ) form the integer sequence

1, 3, 5, 7, 11, 15, 17, 19, 23, 27, 29, 31, 35, 39, 41, ... [30]

S. Naranan, an Indian physicist, seeks a more restricted class of generalized puzzles, with only three terms, and with equal to the least common multiple of the denominators of the three unit fractions, finding only seven possible triples of fractions that meet these conditions. [11]

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. [10]

See also

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References

  1. 1 2 Sesiano, Jacques (2014), "Le partage des chameaux", Récréations Mathématiques au Moyen Âge (in French), Lausanne: Presses Polytechniques et Universitaires Romandes, pp. 198–200, archived from the original on 2023-03-25, retrieved 2023-03-25
  2. 1 2 3 4 5 6 Stockmeyer, Paul K. (September 2013), "Of camels, inheritance, and unit fractions", Math Horizons , 21 (1): 8–11, doi:10.4169/mathhorizons.21.1.8, JSTOR   10.4169/mathhorizons.21.1.8, MR   3313765, S2CID   125145732
  3. 1 2 Ben-Chaim, David; Shalitin, Yechiel; Stupel, Moshe (February 2019), "Historical mathematical problems suitable for classroom activities", The Mathematical Gazette , 103 (556): 12–19, doi:10.1017/mag.2019.2, S2CID   86506133
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  5. ↑ Anne, Premchand (1998), "Egyptian fractions and the inheritance problem", The College Mathematics Journal , 29 (4): 296–300, doi:10.1080/07468342.1998.11973958, JSTOR   2687685, MR   1648474
  6. ↑ Cajori, Florian (1894), A History of Mathematics, MacMillan and Co., pp. 79–80
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  8. ↑ Çarkoğlu, Ali; Erdoğan, Emre (1998), "Fairness in the apportionment of seats in the Turkish legislature: is there room for improvement?", New Perspectives on Turkey, 19: 97–124, doi:10.1017/s0896634600003046, S2CID   148547260
  9. 1 2 3 4 Ageron, Pierre (2013), "Le partage des dix-sept chameaux et autres arithmétiques attributes à l'immam 'Alî: Mouvance et circulation de récits de la tradition musulmane chiite" (PDF), Revue d'histoire des mathématiques (in French), 19 (1): 1–41, archived (PDF) from the original on 2023-03-24, retrieved 2023-03-24; see in particular pp. 13–14.
  10. 1 2 3 4 5 Nascimento, Márcio Luís Ferreira; Barco, Luiz (September 2016), "The man who loved to count and the incredible story of the 35 camels", Journal of Mathematics and the Arts , 10 (1–4): 35–43, doi:10.1080/17513472.2016.1221211, S2CID   54030575, archived from the original on 2023-03-25, retrieved 2023-03-25
  11. 1 2 Naranan, S. (1973), "An "elephantine" equation", Mathematics Magazine , 46 (5): 276–278, doi:10.2307/2688266, JSTOR   2688266, MR   1572070
  12. ↑ Fletcher, James Phillips (1850), Notes from Nineveh: And Travels in Mesopotamia, Assyria and Syria, Lea & Blanchard, p. 206
  13. ↑ Maxham, Ephraim; Wing, Daniel Ripley (24 October 1850), "A Wise Judge", The Eastern Mail, Waterville, Maine, vol. 4, no. 14, p. 3, archived from the original on 2023-03-24, retrieved 2023-03-24
  14. ↑ "Problem", Notes and queries, The Mathematical Monthly, 1 (11): 362, August 1859, archived from the original on 2023-03-25, retrieved 2023-03-25
  15. ↑ Professor Hoffmann (1893), "No. XI—An Unmanageable Legacy", Puzzles Old and New, London: Frederick Warne and Co., p. 147; solution, pp. 191–192
  16. ↑ Fourrey, Émile (1899), "Curieux partages", Récréations arithmétiques (in French), Paris: Librairie Nony, p. 159
  17. ↑ Morrell, E. W. (February 1897), "Problems for solution: arithmetic, no. 76", The American Mathematical Monthly , 4 (2): 61, doi:10.2307/2970050, JSTOR   2970050
  18. ↑ White, William F. (1908), "Puzzle of the camels", A Scrap-Book of Elementary Mathematics: Notes, Recreations, Essays, The Open Court Publishing Company, p. 193
  19. 1 2 Wolff, Sir Henry Drummond (1908), "A Parsee inspiration", Rambling Recollections, vol. II, London: MacMillan and Co., p. 56
  20. ↑ Wentworth, George; Smith, David Eugene (1909), Complete Arithmetic, Wentworth–Smith Mathematical Series, Ginn and Company, p. 467
  21. ↑ Browne, Ray B. (Fall 1961), "Riddles from Tippecanoe County, Indiana", Midwest Folklore , 11 (3): 155–160, JSTOR   4317919
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  24. ↑ Coyle, Stephen (November 2000), "Fractions give me the hump", Mathematics in School, 29 (5): 40, JSTOR   30215451
  25. ↑ Anspach, C. L. (December 1939), "Eternal values", Christian Education, 23 (2): 96–102, JSTOR   41173250
  26. ↑ Chodosh, Hiram E. (March 2008), "The eighteenth camel: mediating mediation reform in India", German Law Journal, 9 (3): 251–283, doi:10.1017/s2071832200006428, S2CID   141042869
  27. ↑ Ost, F. (July 2011), "The twelfth camel, or the economics of justice", Journal of International Dispute Settlement, 2 (2): 333–351, doi:10.1093/jnlids/idr003
  28. ↑ Teubner, Gunther (2001), "Alienating justice: on the social surplus value of the twelfth camel", in Nelken, David; Priban, Jiri (eds.), Law's New Boundaries: Consequences of Lega Autopoiesis, London: Ashgate, pp. 21–44, archived from the original on 2023-07-08, retrieved 2023-03-26
  29. ↑ Swann, W. F. G. (July 1931), "Greetings of the American Physical Society", The Scientific Monthly, 33 (1): 5–10, Bibcode:1931SciMo..33....5S, JSTOR   15070
  30. ↑ Sloane, N. J. A. (ed.), "SequenceA085493(Numbers k having partitions into distinct divisors of k + 1)", The On-Line Encyclopedia of Integer Sequences , OEIS Foundation
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