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This chapter is part of a book that is no longer available to purchase from Cambridge Core

31 - The Three Jug Problem

from Coffee Break 6

Svetoslav Savchev  and
Titu Andreescu
Svetoslav Savchev
Affiliation:
Matematika
Titu Andreescu
Affiliation:
University of Texas at Dallas
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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.

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Mathematical Miniatures , pp. 124 - 127
Publisher: Mathematical Association of America
Print publication year: 2003

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