  We report the results of exact diagonalization studies of Hubbard models on a
$4\times 4$ square lattice with periodic boundary conditions and various
degrees and patterns of inhomogeneity, which are represented by inequivalent
hopping integrals $t$ and $t^{\prime}$. We focus primarily on two patterns, the
checkerboard and the striped cases, for a large range of values of the on-site
repulsion $U$ and doped hole concentration, $x$. We present evidence that
superconductivity is strongest for $U$ of order the bandwidth, and intermediate
inhomogeneity, $0 <t^\prime< t$. The maximum value of the ``pair-binding
energy'' we have found with purely repulsive interactions is $\Delta_{pb} =
0.32t$ for the checkerboard Hubbard model with $U=8t$ and $t^\prime = 0.5t$.
Moreover, for near optimal values, our results are insensitive to changes in
boundary conditions, suggesting that the correlation length is sufficiently
short that finite size effects are already unimportant.
