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We present the first systematic algorithm to estimate the maximum packingdensity of spheres when the grain sizes are drawn from an arbitrary sizedistribution. With an Apollonian filling rule, we implement our technique fordisks in 2d and spheres in 3d. As expected, the densest packing is achievedwith power-law size distributions. We also test the method on homogeneous andon empirical real distributions, and we propose a scheme to obtainexperimentally accessible distributions of grain sizes with low porosity. Ourmethod should be helpful in the development of ultra-strong ceramics and highperformance concrete.

How dense can one pack spheres of arbitrary size distribution?