13. In a regular tetrahedron, six planes are drawn through each edge and the midpoint of its opposite edge. Determine how many parts these planes divide the tetrahedron into. Also, when the volume of the tetrahedron is 1, find the volume of each part.
(1974 GDR Competition Problem)
13. Since any plane drawn contains one of the line segments connecting the midpoints of opposite edges of the tetrahedron, all these planes pass through the intersection point of these line segments, which is inside the tetrahedron (a property of the centroid). Therefore, the 6 planes divide the entire space into polyhedral angles with a common vertex. This indicates that any of the smaller tetrahedra must have one face belonging to one of the faces of the original tetrahedron. On the other hand, two faces of the original tetrahedron cannot be two faces of a smaller tetrahedron. This is because the two faces are necessarily separated by the plane passing through their common edge. Thus, the number of smaller tetrahedra is equal to the number of pieces into which the surface of the original tetrahedron is divided. Since each face of the original tetrahedron is divided into 6 pieces (by its medians), the total number of pieces is \(6 \times 4 = 24\). Considering the symmetry with respect to any of the six planes, it can be seen that all the smaller tetrahedra are equal, and thus each has a volume of \(\frac{1}{24}\).