In dimension \(d\), how many non-overlapping unit spheres can simultaneously touch a central unit sphere? The answer is known exactly only for \(d \in \{1, 2, 3, 4, 8, 24\}\) — respectively \(2, 6, 12, 24, 240\) and \(196960\) — and is open in every other dimension, including \(d = 5\).
Born in the famous 1694 dispute between Newton, who held that twelve spheres could touch a central one, and Gregory, who thought thirteen might fit: Newton was right, though a rigorous proof waited until Schütte and van der Waerden (1953). The exact values in dimensions 8 and 24 come from Delsarte's linear-programming bounds of the 1970s; dimension 4 fell to Musin (2003). The spectacular 2016–2017 packing breakthroughs of Viazovska and collaborators settled optimal density in 8 and 24 dimensions, a neighbouring triumph that left the remaining kissing numbers exactly where they were.
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