I suggest you do a search for "sphere packing problem" and see the variety of problems that fall under this category. I ran into several before hitting the "kissing number" problem for idential spheres in different numbers of dimensions that you seem to think is the only one.
From what I read, the original problem was popularized by Kepler when he guessed how tightly you could pack identical spheres in 3 dimensions. This one was recently solved (by proving the "obvious").
(but my friends call me "Tye")
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