  A 2-coloring of the n-cube in the n-dimensional Euclidean space can be
considered as an assignment of weights of 1 or 0 to the vertices. Such a
colored n-cube is said to be balanced if its center of mass coincides with its
geometric center. Let $B_{n,2k}$ be the number of balanced 2-colorings of the
n-cube with 2k vertices having weight 1. Palmer, Read and Robinson conjectured
that for $n\geq 1$, the sequence $\{B_{n,2k}\}_{k=0, 1 ... 2^{n-1}}$ is
symmetric and unimodal. We give a proof of this conjecture. We also propose a
conjecture on the log-concavity of $B_{n,2k}$ for fixed k, and by probabilistic
method we show that it holds when n is sufficiently large.
