Say you have a standard solid, rigid sphere like a ball
bearing. Surround that sphere with as many
identical spheres as you possibly can. Hint:
the maximum number of equal sized spheres that you can put around a single sphere
is 12, according to sphere packing geometry.
You can see how that works here, if you imagine removing the
top orange and adding three oranges below, so you have three above, three below
and six surrounding the central orange in the middle, for a total of twelve.
Call this the first layer, or Layer 1, and then keep adding more
layers.
How many spheres in total will you have when you reach Layer
100? For bonus points, how many spheres
will there be in Layer 100?
(And for extra extra points, is there a formula to calculate the number of spheres with N layers that is more than the just the summation of all spheres in all the layers plus one [for the first sphere]?)
(And for extra extra points, is there a formula to calculate the number of spheres with N layers that is more than the just the summation of all spheres in all the layers plus one [for the first sphere]?)
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