Mathematical Recreations in the Middle Ages

Jacques Sesiano taught history of mathematics at the EPFL (Swiss Federal Institute of Technology) in Lausanne.

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• Chapter I. Sharing liquid by decanting1. First appearance 2. Decanting operations3. Other examples4. Solving conditions5. GeneralizationChapter II. Sharing liquid and vessels1. First appearance and solution2. Further examples3. Solving conditions according to Bachet de Méziriac4. GeneralizationChapter III. Minimal set of weights1. Introduction2. Arabic references3. Finding the solution4. Mediaeval examples5. GeneralizationChapter IV. Successive distributions1. Doorkeepers2. Giving to the church3. The last willChapter V. Mutual borrowing1. Moschos’ problem 2. Findingapurse3. Buyingahorse 4. The dishonest partners5. Examples Chapter VI. Filling and emptying cisterns1. Mediaeval examples2. Related problems Chapter VII. Messengers1. Introduction2. Progress of a single person3. Pursuit4. Meeting 5. Dog and hare6. Moving back and forthChapter VIII. Large numbers1. First occurrences2. Selected mediaeval examples3. Doubling on the chessboard4. Earth’s size, ancient and mediaeval estimates5. Some illustrations of the result of doubling 6. Estimating the earth’s weight 7. Perfect numbers Chapter IX. Arrangements1. Musical chairs 2. The broken eggs 3. Unwanted passengers 4. The nuns’ cells Chapter X. Crossing a river1. Goat, wolf and cabbages 2. Two couples and a boat with two places3. Three couples and a boat with two places 4. Four couples and a boat with two places 5. Five or six couples and a boat with two places 6. Four couples and a boat with three places7. Five couples and a boat with three places Chapter XI. Miscellanea1. Sharing with a traveller 2. Multiplication of rabbits 3. The dishonest servant4. The future heirs 5. Sharing camels 6. Striking the hour7. Two workers, same work 8. Same earnings from same items in different quantities Chapter XII. Family relationships1. Two men marry each other’s sister 2. Two men marry each other’s mother 3. Two men marry each other’s daughter 4. Of two men, one marries the other’s mother and the second the other’s daughter 5. Father and son marry mother and daughter 6. Father and son marry daughter and mother 7. Three generationsChapter XIII. The knight’s tour1. Euler’s study 2. Mediaeval toursChapter XIV. Magic squares1. Early history 2. Arrival of magic squares in late mediaeval Europe 3. First attempts at reconstruction Chapter XV. Infinite sets1. Natural numbers and rational fractions 2. Finite and infinite in Greece and the Middle Ages 3. Examples of comparing infinites in the Middle Ages 4. Ancient and mediaeval root approximations Chapter XVI. Geometrical recreations1. The tower and the river 2. Measuring an inaccessible height3. The ladder leaning against a wall 4. The two towers 5. Two towers and a fountain6. Further problem of two towers 7. The falling tree 8. The stone thrown into a cisternChapter XVII. Hidden numbers1. The walnuts2. The novenary3. The unknown date4. The unknown sum 5. The hidden ring 6. Two people and two things 7. Three people and three things8. The game with coinsAppendicesA. Brief outline of the history of mathematics1. Arithmetic and its applications 2. Progressions and series.3. Algebra4. Symbolism