Equilibrium thermodynamic phenomena such as the Maxwell–Boltzmann distribution of molecular velocities are rare in systems of macroscopic particles interacting by mechanical collisions. This paper reports a system composed of millimeter-sized polymer objects that under mechanical agitation exhibits a “discretization” of the configurations of the system, and has a distribution of the probabilities of these configurations that is analogous to a Boltzmann distribution. The system is composed of spheres and a three-link chain on a bounded horizontal surface, shaken with an aperiodic but not completely random horizontal motion. Experiments were performed at different strengths of agitation (quantified by the frequency of agitation, f, at constant amplitude) and densities of spheres (quantified by the filling ratio, FR). The chain was typically found in one of three conformations – extended, single folded, and double folded – because, under collisions with the spheres, adjacent links were stable mechanically only when fully extended or fully folded. The probabilities of the different conformations of the chain could be described by a Boltzmann distribution in which the “temperature” depended on f and the “energies” of conformations on FR. The predictions of the Boltzmann formula using empirically determined “temperatures” and “energies” agreed with measurements within two experimental standard deviations in 47 out of 48 experiments.