Recreational Problems in Medieval Mathematics - Conclusions and Questions

Finally, we should note that the purse problem as well as the horse problem turn up in other places as well, from India to early modern Europe. The question that is difficult to answer, however, is how did these problems travel?

There is no direct evidence to answer this question. We see that frequently the problems are essentially identical in works separated by time and distance – albeit sometimes with different constants.  Yet the solution methods have considerable variation. Some authors attack these problems in what we would call an algebraic fashion, while others use more arithmetic methods. Sometimes the problems are abstract numerical problems, while at other times they are tied to a “story”. It is difficult to believe, however, that each of these people invented the problem from scratch, since whether one is in the abstract or story mode, the problem seems too specialized for independent invention.  Thus, somehow the idea of the problems, at least, must have traveled. It is certainly possible to imagine travelers between east and west carrying knowledge of these problems, perhaps even without the solution. We know, for example, that Diophantus’s Arithmetica was translated into Arabic by the tenth century, so would have been available for al-Karaji and later writers in Arabic. However, we know of no translation of Diophantus into Hebrew in the Middle Ages, so perhaps Levi ben Gershon learned of this through Arabic sources. On the other hand, we have no direct evidence of Indian problems being taken to western Europe, although, of course, the Indian decimal place-value system did make this journey. But given that Mahāvīra himself did not have an accurate idea of the solution procedure of the problem, it would seem that this problem must have come to him from elsewhere – whether from India or some other country. So there are lots of puzzles out there that these puzzle problems engender. It seems that we will have to wait for more research to get good answers.

About the Author

Victor J. Katz is Professor Emeritus of Mathematics of the University of the District of Columbia; founding co-editor (with Frank J. Swetz) of MAA Convergence; author of History of Mathematics: An Introduction (3rd ed., 2009), widely recognized as the definitive general history of mathematics for professionals, instructors, and students; co-author with Karen Parshall of Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century (2014); and editor of two sourcebooks for history of mathematics, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (2007) and Sourcebook in the Mathematics of Medieval Europe and North Africa (2016). – Janet Beery, Editor, MAA Convergence