Mathematical Recreations in the Middle Ages

Chapter I. Sharing liquid by decanting 1. First appearance 2. Decanting operations 3. Other examples 4. Solving conditions 5. Generalization Chapter II. Sharing liquid and vessels 1. First appearance and solution 2. Further examples 3. Solving conditions according to Bachet de Méziriac 4. Generalization Chapter III. Minimal set of weights 1. Introduction 2. Arabic references 3. Finding the solution 4. Mediaeval examples 5. Generalization Chapter IV. Successive distributions 1. Doorkeepers 2. Giving to the church 3. The last will Chapter V. Mutual borrowing 1. Moschos’ problem 2. Findingapurse 3. Buyingahorse 4. The dishonest partners 5. Examples Chapter VI. Filling and emptying cisterns 1. Mediaeval examples 2. Related problems Chapter VII. Messengers 1. Introduction 2. Progress of a single person 3. Pursuit 4. Meeting 5. Dog and hare 6. Moving back and forth Chapter VIII. Large numbers 1. First occurrences 2. Selected mediaeval examples 3. Doubling on the chessboard 4. Earth’s size, ancient and mediaeval estimates 5. Some illustrations of the result of doubling 6. Estimating the earth’s weight 7. Perfect numbers Chapter IX. Arrangements 1. Musical chairs 2. The broken eggs 3. Unwanted passengers 4. The nuns’ cells Chapter X. Crossing a river 1. Goat, wolf and cabbages 2. Two couples and a boat with two places 3. 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 places 7. Five couples and a boat with three places Chapter XI. Miscellanea 1. Sharing with a traveller 2. Multiplication of rabbits 3. The dishonest servant 4. The future heirs 5. Sharing camels 6. Striking the hour 7. Two workers, same work 8. Same earnings from same items in different quantities Chapter XII. Family relationships 1. 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 generations Chapter XIII. The knight’s tour 1. Euler’s study 2. Mediaeval tours Chapter XIV. Magic squares 1. Early history 2. Arrival of magic squares in late mediaeval Europe 3. First attempts at reconstruction Chapter XV. Infinite sets 1. 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 recreations 1. The tower and the river 2. Measuring an inaccessible height 3. The ladder leaning against a wall 4. The two towers 5. Two towers and a fountain 6. Further problem of two towers 7. The falling tree 8. The stone thrown into a cistern Chapter XVII. Hidden numbers 1. The walnuts 2. The novenary 3. The unknown date 4. The unknown sum 5. The hidden ring 6. Two people and two things 7. Three people and three things 8. The game with coins Appendices A. Brief outline of the history of mathematics 1. Arithmetic and its applications 2. Progressions and series. 3. Algebra 4. Symbolism