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.