We define the mechanical complexity C(P) of a 3-dimensional convex polyhedron P , interpreted as a homogeneous solid, as the difference between the total number of its faces, edges and vertices and the number of its static equilibria; and the mechanical complexity C(S, U) of primary equilibrium classes (S, U) E with S stable and U unstable equilibria as the infimum of the mechanical complexity of all polyhedra in that class. We prove that the mechanical complexity of a class (S, U) E with S, U > 1 is the minimum of 2(f + v − S − U) over all polyhedral pairs (f, v), where a pair of integers is called a polyhedral * The authors acknowledge the support of the NKFIH Hungarian Research Fund grant 119245 and of grant BME FIKP-VÍZ by EMMI. The authors thank Mr. Otto Albrecht for backing the prize for the complexity of the Gömböc-class. Any solution should be sent to the corresponding author as an accepted publication in a mathematics journal of worldwide reputation and it must also have general acceptance in the mathematics community two years after. The authors are indebted to Dr. Norbert Krisztián Kovács for his invaluable advice and help in printing the 9 tetrahedra and 7 pentahedra, and the two referees for their valuable comments and one of them for posing Problems 5.2 and 5.4. † Corresponding author. The author has been supported by the Bolyai Fellowship of the Hungarian Academy of Sciences and partially supported by the UNKP-19-4 New National Excellence Program of the Ministry of Human Capacities.