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Geoboards, polygons and Pick’s Theorem

Published online by Cambridge University Press:  01 August 2016

Alan F. Beardon
Alan F. Beardon*
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB
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Extract

A geoboard is a flat piece of wood with nails driven in at integer lattice points, and polygons are constructed on the geoboard by placing rubber bands over the nails. At a workshop for teachers, held at Stellenbosch in South Africa, and organised by the African Institute for Mathematical Sciences Schools Enrichment Centre (see http://aims.ac.za/aimssec), it was conjectured that if two polygons on a geoboard overlap, then the the area of the overlap is rational. It is not difficult to prove this but, as we shall see, the ideas in the proof go far beyond the original problem and may be of interest to some Gazette readers.

Type
Articles
Information
The Mathematical Gazette , Volume 93 , Issue 527 , July 2009 , pp. 228 - 231
Copyright
Copyright © The Mathematical Association 2009

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References

1. Reeve, J.E., On the volume of lattice polyhedra, Proc. London Math. Soc. (3) 7 (1957) pp. 378–395.CrossRefGoogle Scholar
2. Ehrhart, E., Sur un problème de géométrie diophantienne linéare II, J. reine angewandte Math. 227 (1967) pp. 25–49.Google Scholar