Mathematical Recreations in the Middle Ages

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

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