In this paper, we examine the effect of dissecting an n-dimensional simplex using cevians (cross-sections passing through n&1 of the vertices of the simplex). We describe a formula for the number of pieces the simplex is dissected into using a polynomial involving only the number of each type of cevian. The polynomial in question involves terms involving the edges of the simplex, but discarding those terms involving cycles of the underlying graph. Thus, we call such a polynomial a ``forest polynomial.'' 1999 Academic Press A cevian of a triangle T is a line segment connecting a point on an edge of T to the vertex opposite this edge. To extend this definition for higher dimensional simplices, we define a cevian of a non-degenerate simplex S to be a simplex joining a point on an edge of S to the remaining vertices of S. Alternatively, we may view a cevian as a cross-section of S which passes through all but two of its vertices. By this definition, a cevian of a tetrahedron, as depicted in Fig. 1, is a triangle sharing two vertices of the tetrahedron and having its third vertex on the edge joining the other two vertices of the tetrahedron.
We wish to examine and later generalize to higher dimensions the answer to the following question: Question 1. Given triangle T with vertices A, B, and C, suppose we draw a cevians from vertex A, b cevians from B, and c cevians from C so that no three cevians intersect in one point in the interior of T. Into how many pieces is T dissected?
The following solution was given by a fifth-grade student (see [1] for details). Suppose the solution depends only on a, b, and c, and not on the location of the cevians. Then draw a new picture with the cevians ``interfering'' as little as possible as depicted in Fig. 2.