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Let $C(n)$ be the solution to the contact number problem, i.e., the maximumnumber of touching pairs among any packing of $n$ congruent spheres in$\mathbb{R}^3$. We prove the long conjectured values of $C(6)=12, C(7)=15$, and$C(8)=18$. The proof strategy generalizes under an extensive case analysis to$C(9)=21, C(10) = 25, C(11) = 29, C(12) = 33$, and $C(13) = 36$. These resultshave great importance for condensed matter physics, materials science,crystallography, organic and physical chemistry of interfaces.

On Arrangements of Six, Seven, and Eight Spheres: Maximal Bonding of Monatomic Ionic Compounds