Book contents
- Frontmatter
- Preface
- Contents
- Warm-up Problem Set
- Coffee Break 1
- Coffee Break 2
- Coffee Break 3
- Coffee Break 4
- Coffee Break 5
- Coffee Break 6
- 31 The Three Jug Problem
- 32 Rectifying Trajectories
- 33 Numerical Systems
- 34 More on Polynomials
- 35 Geometric Transformations
- Coffee Break 7
- Coffee Break 8
- Coffee Break 9
- Instead of an Afterword
- Glossary
- About the Authors
31 - The Three Jug Problem
from Coffee Break 6
- Frontmatter
- Preface
- Contents
- Warm-up Problem Set
- Coffee Break 1
- Coffee Break 2
- Coffee Break 3
- Coffee Break 4
- Coffee Break 5
- Coffee Break 6
- 31 The Three Jug Problem
- 32 Rectifying Trajectories
- 33 Numerical Systems
- 34 More on Polynomials
- 35 Geometric Transformations
- Coffee Break 7
- Coffee Break 8
- Coffee Break 9
- Instead of an Afterword
- Glossary
- About the Authors
Summary
You can find this same title in the famous book Geometry Revisited by Coxeter and Greitzer. But the story we are going to tell here is quite different. It is about the 1993 Putnam problem B-6:
Three nonnegative integers are given. We may choose two of them, say x and y, and if x ≤ y, replace them by 2x and y − x. Prove that, after a finite number of such operations, it is possible to obtain 0.
There is nothing wrong with the statement of the problem but we like another one better:
Three jugs are given with water in them, each containing an integer number of pints. It is allowed to pour into any jug as much water as it already contains, from any other jug. Prove that after several such pourings it is possible to empty one of the jugs. (Assume that the jugs are sufficiently large; each of them can contain all the water available.)
This problem was posed for the 1971 All-Union Olympiad. It originated in a paper by the prominent Russian algebraist Alexei Shirshov, where he used it for purposes far beyond the scope of our story. The late Shirshov was a nontraditional mathematician with a highly nontraditional career.
- Type
- Chapter
- Information
- Mathematical Miniatures , pp. 124 - 127Publisher: Mathematical Association of AmericaPrint publication year: 2003