In this presentation, I show that eigenenergies and eigenstates of a system consisting of four one-dimensional hard-core particles with masses 6m, 2m, m, and 3m in a hard-wall box can be found exactly using Bethe ansatz. The ansatz is based on the exceptional affine reflection group $F_4$ associated with the symmetries and tiling properties of an octacube--a Platonic solid unique to four dimensions, with no three-dimensional analogues. The construction we use can be extended to any reflection group--affine or finite--whose Coxeter diagram does not have bifurcations.