Yet more frogs

Extending a recent paper by Derek Holton, we show how to represent the algorithm for the Frog Problem diagrammatically. This diagrammatic representation suggests a simpler proof of the symmetrical case (equal numbers of frogs of each colour) by allowing the even and odd cases to be treated together....

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Bibliographic Details
Published in:International journal of mathematical education in science and technology 2011-06, Vol.42 (4), p.524-533
Main Author: Shutler, Paul M.E.
Format: Article
Language:English
Subjects:
Algorithms
Color
combinatorics
Frogs
induction
Mathematical Concepts
Mathematical Logic
Mathematics Education
Mathematics Instruction
Problem Solving
Studies
the Frog Problem
Validity
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Summary:Extending a recent paper by Derek Holton, we show how to represent the algorithm for the Frog Problem diagrammatically. This diagrammatic representation suggests a simpler proof of the symmetrical case (equal numbers of frogs of each colour) by allowing the even and odd cases to be treated together. It also provides a proof in the asymmetrical case (unequal numbers of frogs) as an extension of the symmetrical case. The issue of whether frogs of a given colour should be allowed to move in either direction is discussed. While it is possible to restrict to the case of movement in a single direction, results for bi-directional movement can be obtained by making the correspondence between the algorithm and its diagrammatic representation more concrete. The Frog Problem then becomes a form of constrained shortest path problem around the diagram, and from this point of view optimality of the algorithm becomes much clearer.
ISSN:0020-739X
1464-5211
DOI:10.1080/0020739X.2010.543164