How algebra spoiled recreational problems: A case study in the cross-cultural dissemination of mathematics

Abstract

This paper deals with a sub-class of recreational problems which are solved by a simple memorized rule resulting from an elementary arithmetical or algebraic solution, called proto-algebraic rules. Their recreational aspect is derived from a surprise or trick solution which is not immediately obvious to the subjects involved. Around 1560 many such problems wane from arithmetic and algebra textbooks to reappear in the eighteenth century. Several hypotheses are investigated why popular Renaissance recreational problems lost their appeal. We arrive at the conclusion that the emergence of algebra as a general problem solving method changed the scope of what is considered recreational in mathematics. SommarioQuesto saggio tratta di una sottoclasse di problemi ricreativi risolti tramite memorizzazione di una semplice regola risultante da una soluzione algebraica o aritmetica, chiamata regola proto-algebraica. L'aspetto ricreativo di questi problemi deriva da una soluzione a sorpresa o da un trucco non immediatamente ovvi ai soggetti coinvolti. Intorno al 1560 svariati problemi di questo tipo sparirono dai manuali di algebra e aritmetica, per riapparire nel diciottesimo secolo. Diverse ipotesi sono vagliate sul perché problemi ricreativi popolari nel Rinascimento persero attrattività, per giungere alla conclusione che l'emergere dell'algebra come metodo generale di risoluzione di problemi cambiò la portata di ciò che era considerato ricreativo in matematica.

"A difficult case": Pacioli and Cardano on the Chinese Rings

“…Such problems travel easily across cultural boundaries along the trade routes while embedding mathematical practices which become adapted to a new cultural context [15]. With the Chinese rings puzzle, we also have the material culture of crafting, selling, transporting and using the contrivance.…”

“…It is covered in all the classic works such as [22,, [3,[80][81][82][83][84][85], [8,Problem 417] and [12,[15][16][17]. In French nineteenth-century works ( [22], [13]), the puzzle is also known by the name 'Baguenaudier'.…”

Recreational Mathematics – Only For Fun?

“…Apart from understanding mathematical knowledge, in the Chinese curriculum (for age [6][7][8][9][10][11][12][13][14] [19] we can read that children should also display progress and development in the affective areas such as attitudes, emotions, and values. This strand is called Affection and Attitudes and students are supposed to "[p]ossess some degree of creative spirits and practical abilities, and develop sufficiently in areas of general abilities, affection and attitudes" [19, page 10].…”

“…Dudley cited in [30] establishes that recreational mathematics goes back to Babylonian tablets and The Rhind papyrus, making this an activity more than 3500 years old. Some problems from these historical texts are presented in [13,30] as examples. Heefer [13] concludes that many of the same kinds of problems appear in very different cultures and areas, illustrating this with the sliding ladder problem, where a ladder of a given length is leaning against a wall and slides from its original position.…”

“…Some problems from these historical texts are presented in [13,30] as examples. Heefer [13] concludes that many of the same kinds of problems appear in very different cultures and areas, illustrating this with the sliding ladder problem, where a ladder of a given length is leaning against a wall and slides from its original position. The fact we could find this problem in the old Chinese mathematical texts, Old-Babylonian tablets, Sanskrit mathematical texts and so on (cf.…”