Further Results About Odd Equidissection

Let P be a simply connected polygon in the rry-plane with an oriented boundary. Orient the edges on the boundary of P to be consistent with the given orientation. Two edges of P are said to be equivalent if they are parallel. If the sum of the vectors in each equivalence class is the 0-vector, we call P a special polygon. In 2000 a general conjecture was proposed by Stein: a special polygon cannot be cut into an odd number of triangles of equal areas. It has been proved by means of Sperner's lemma and p?adic valuation that the conjecture holds for polygons with at most six sides. In this paper we prove the existence of special n-polygon for any integer n > 6 and that the conjecture holds for some typical types of special polygons with seven sides.