We report the results of exact diagonalization studies of Hubbard models on a $4times 4$ 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.
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