The stability of an assemblage of contacting rigid bodies without fric
tion is investigated, A method is presented for finding an orientation
of the assembly so that the assembly remains motionless under gravity
. If no stable orientation exists for an assembly, the method finds th
e ''least'' unstable orientation. The metric used to measure stability
is based on the second time-rate of change of the gravitational poten
tial energy, and the desired orientation for an assembly is expressed
in terms of an optimization problem involving changes in potential ene
rgy. The problem of finding stable or maximally-stable orientations is
formulated as a constrained maximin problem. The maximin problem is s
hown to be a variant of standard zero-sum matrix games, and can be sol
ved using linear programming. The method is the first general method f
or automatically determining stable orientations. Example assemblies a
re presented.