The goal of this paper is to tell the hitherto known history of an old question concerning the so-called recreative mathematics: this application of arithmetic techniques to every-day situations has surprisingly spread in Asia and Europe through the centuries, and it shows unforeseen connections between remote places, times, and cultures. The presence of a similar or identical question in different contexts can both help historians of mathematics to find unexplored links and dependences between scholars and mathematical discoveries in geographically and culturally far environments, and provide to tout-court historians and cultural anthropologists brand-new material sources to investigate daily life and civilization streams. The way this passage happened, is often a mystery hard to explore, but, from these hints, scholars can rightly be sure that such links existed: in fact recreative or, better, applied mathematics has always and everywhere been considered a minor branch of this discipline, devoted, as it was, to education or game, so that it has never been censored for any intellectual or religious reason and no filters of dominant culture have been applied. We are considering a problem concerning indeterminate arithmetic, that has been posed in slightly different terms (depending on historical and cultural environment) by famous or anonymous scholars. According to a legend, during 3rd century B.C.E. the problem was first solved in China by General Han Xin of Emperor Liu Bang in order to count up his soldiers: he ordered them to form groups of equal size and considered how many men were left over. After some groupings using different set dimensions, he found out the exact number. Chronologically speaking, the first written appearance of the problem dates back to the 3rd century C.E. in ancient China, where Sunzi Suan Jing, in his mathematical handbook (Sun Tzu Suan Ching - Master Sun's Mathematical Manual), posed the question (#26, known as “problem of Master Sun”). Then, the problem is found in India in the most important arithmetical work of the 7th century, Brahmasphutasiddanta (The Opening of the Universe), by the astronomer and mathematician Brahmagupta, and at this turn it deals with an amount of eggs in a basket; in the same years the problem is presented in 26 different examples in Bhaskara I’s astronomical work Maha-Bhaskariya (Great Book of Bhaskara). Through Indian mediation, the problem (and Indian numeral system with it) passed to Arabic culture: in the10th century, it was put in al-Baghdadi’s al-Takmila, a treatise upon different systems of arithmetic (i.e. counting on the fingers, sexagesimal system, Indian numerals and fractions), and was posed again but in an abstract way. During the 13th century, we see a double emergence of the problem: in Italy, where in Leonardo Fibonacci’s Liber Abaci (Book on Calculation) two abstract examples are found; and in China, where it was solved with Ta-Yen rule, a method created by Qin Jiushao, described in his Shushu jiuzhang (Mathematical Treatise in Nine Sections). Later we have examples in different European treatises of the Renaissance: in France, in Chuquet’s Triparty en la science des nombres (Book on calculation in three sections), the problem deals another time with eggs; in Germany, Regiomontanus’ Collectanea mathematica, an appendix of the Latin translation (done by Gerard of Cremona) of al-Jabr w’al-Muqabalah by al-Khawarizmi, contains another abstract example.

Recreative mathematics : soldiers, eggs and a pirate crew / N. Ambrosetti - In: Mathknow : mathematics, applied sciences and real life / [a cura di] M. Emmer, A.M. Quarteroni. - New York : Springer, 2009 Apr. - ISBN 978-88-470-1121-2. - pp. 183-192 (( convegno Mathknow08 : mathematics, applied sciences and real life tenutosi a Milano nel 2008.

Recreative mathematics : soldiers, eggs and a pirate crew

N. Ambrosetti
Primo
2009

Abstract

The goal of this paper is to tell the hitherto known history of an old question concerning the so-called recreative mathematics: this application of arithmetic techniques to every-day situations has surprisingly spread in Asia and Europe through the centuries, and it shows unforeseen connections between remote places, times, and cultures. The presence of a similar or identical question in different contexts can both help historians of mathematics to find unexplored links and dependences between scholars and mathematical discoveries in geographically and culturally far environments, and provide to tout-court historians and cultural anthropologists brand-new material sources to investigate daily life and civilization streams. The way this passage happened, is often a mystery hard to explore, but, from these hints, scholars can rightly be sure that such links existed: in fact recreative or, better, applied mathematics has always and everywhere been considered a minor branch of this discipline, devoted, as it was, to education or game, so that it has never been censored for any intellectual or religious reason and no filters of dominant culture have been applied. We are considering a problem concerning indeterminate arithmetic, that has been posed in slightly different terms (depending on historical and cultural environment) by famous or anonymous scholars. According to a legend, during 3rd century B.C.E. the problem was first solved in China by General Han Xin of Emperor Liu Bang in order to count up his soldiers: he ordered them to form groups of equal size and considered how many men were left over. After some groupings using different set dimensions, he found out the exact number. Chronologically speaking, the first written appearance of the problem dates back to the 3rd century C.E. in ancient China, where Sunzi Suan Jing, in his mathematical handbook (Sun Tzu Suan Ching - Master Sun's Mathematical Manual), posed the question (#26, known as “problem of Master Sun”). Then, the problem is found in India in the most important arithmetical work of the 7th century, Brahmasphutasiddanta (The Opening of the Universe), by the astronomer and mathematician Brahmagupta, and at this turn it deals with an amount of eggs in a basket; in the same years the problem is presented in 26 different examples in Bhaskara I’s astronomical work Maha-Bhaskariya (Great Book of Bhaskara). Through Indian mediation, the problem (and Indian numeral system with it) passed to Arabic culture: in the10th century, it was put in al-Baghdadi’s al-Takmila, a treatise upon different systems of arithmetic (i.e. counting on the fingers, sexagesimal system, Indian numerals and fractions), and was posed again but in an abstract way. During the 13th century, we see a double emergence of the problem: in Italy, where in Leonardo Fibonacci’s Liber Abaci (Book on Calculation) two abstract examples are found; and in China, where it was solved with Ta-Yen rule, a method created by Qin Jiushao, described in his Shushu jiuzhang (Mathematical Treatise in Nine Sections). Later we have examples in different European treatises of the Renaissance: in France, in Chuquet’s Triparty en la science des nombres (Book on calculation in three sections), the problem deals another time with eggs; in Germany, Regiomontanus’ Collectanea mathematica, an appendix of the Latin translation (done by Gerard of Cremona) of al-Jabr w’al-Muqabalah by al-Khawarizmi, contains another abstract example.
remainders problem ; mathematics ; Fibonacci ; Regiomontanus ; Chuquet ; Brahmagupta ; China ; Sun Tzu
apr-2009
Politecnico di Milano
Fondazione CARIPLO
Springer
Ufficio Scolastico per la Lombardia
Book Part (author)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/56549
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