A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently s
ymmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.
