  We examine inhomogeneous primordial nucleosynthesis for {\it arbitrary}
distributions $f$ of the baryon-to-photon ratio $\eta$, in the limit where
neither particle diffusion nor gravitational collapse is important. By
discretizing $f(\eta)$ and using linear programming, we show that for a set of
$m$ constraints on the primordial element abundances, the maximum and minimum
possible values of $\bar \eta$ (the final mean value of $\eta$) are given when
$f(\eta)$ is a sum of at most $m+1$ delta functions. Our linear programming
results indicate that when $f$ is taken to be an arbitrary function, there is
no lower bound on $\bar \eta$, while the upper bound is essentially the
homogeneous upper bound.
