Congruent Dudeney Dissections of Polygons I: All the Hinge Points are Vertices of the Polygon

Let α and β be polygons with the same area. A Dudeney dissection of α to β is a partition of α into parts which can be reassembled to produce β as follows: Hinge the parts of α like a string along the perimeter of α, then fix one of the parts to from β with the perimeter of α going into its interior and with its perimeter consisting of the dissection lines in the interior of α, without turning the surfaces over. In this paper we discuss a special case of Dudeney dissection where α is congruent to β, in particular, when all hinge points are on the vertices of the polygon α. We determine necessary and sufficient conditions under which such dissections exist.