How many spheres of radius 10 can fit in a sphere of radius 100?
This begs the question of how exactly you pack in the spheres. Different arrangements take up different amounts of space. Assume you use a “close-packed” pattern where spheres on the same plane are surrounded by 6 neighboring spheres, and you stack these horizontal layers of spheres on top of each other. Consider how many of these you could put on a table, packed into a circle of radius 100. Bascially, you’ll end up with spheres stacked to look like a giant hexagon, like this: http://mathworld.wolfram.com/HexNumber.h… The number of individual spheres in a pack like this is 3n^2 + 3n + 1, where 2n+1 is the number of spheres you have across the middle row. We can’t have 10 because it has to be an odd number. So if 2n+1=9, then n=4 and the count is 3(4)^2 + 3(4) + 1 = 61. You could nest another hexagon above and below this that’s a little smaller, then add up the results. This would still give you some space on the s
