  In this paper we study the viability of persuing analytic variational
techniques for the calculation of glueball masses in 3+1 dimensional
Hamiltonian lattice gauge theory (LGT) in the pure gauge sector. We discuss the
major problems presented by a move from 2+1 to 3+1 dimensions and develop
analytic techniques to approximate the integrals appearing in 3+1 dimensional
variational glueball mass calculations. We calculate $0^{++}$ and $1^{+-}$
glueball masses on a lattice consisting of a single cube. Despite the use of a
very simplistic model, promising signs of an approach to asymptotic scaling is
displayed by the SU(N) $1^{+-}$ glueball mass as N is increased.
