It is known how to choose initial data for Einstein's equations describing an arbitrary number of black holes at a moment of time symmetry. This idea has been used to give insight into the cosmological averaging problem. We study the local curvature of the initial data space, for configurations of 8, 120, or 600 black holes obtained by choosing points either regularly or randomly on a 3-sphere. We conclude that the asymptotic regions are remarkably close to that of Schwarzschild, while the region in between shows interesting behaviour. The cosmological back reaction as defined in the recent literature is actually a bit smaller for the random configurations.