We all have seen different versions of the popular children’s toy called ‘stand up kid’. These figures are easy to make as they are loaded figures which have only one stable equilibrium. Such bodies are called monostatic . The problem gets more interesting if one wants to make convex `stand up kids’ using homogeneous material. A recent construction of G.Domokos and P.Varkonyi amazed people and thus generated lot of media attention. Mathematically speaking they answered a question of V. Arnold by constructing a homogeneous, convex body (called Gomboc) which has exactly one stable equilibrium, exactly one unstable equilibrium and does not have any saddle type equilibrium.

From the very beginning the problem of finding a monostatic polyhedron with the smallest number of faces seemed to be of special interest. J.Conway and R.Guy (1969) constructed a monostatic polyhedron with 19 faces. It was long believed that 19 is the smallest such face number. In this talk, with a different idea, we construct a monostatic polyhedron which has only 18 faces. We also consider skeletal versions of the original stability problem. Among others we prove that if the 1, 2 and 3 skeletons of a tetrahedron is constructed of homogeneous