Suppose there is a large sphere of radius $R$. We want to pack it with smaller spheres. The volume of the smaller spheres change depending on where they are situated in the larger sphere. A smaller sphere has the largest volume when it is at the edge of the large sphere as the smaller sphere moves closer to the center of the larger sphere the radius of the smaller spheres shrinks by a quadratic law. When a small sphere is at the edge of the large sphere its radius is $r$. How many small spheres can exist in the larger sphere?

Edit: An example of the quadratic law is $$r(x)=\frac{r}{(R-r-x+1)^2},$$ where $x$ is the distance from center of a small sphere to the center of the large sphere.

I would attempt to solve it by first packing the largest possible small sphere along the border of the large sphere and then draw a sphere tangent to this first layer of small spheres and fill that with the next layer of smaller spheres and so on till the left out space can not handle the smallest possible smaller sphere which has radius $\frac{r}{R-r+1}$.

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    $\begingroup$ What are your thoughts on this? $\endgroup$ – abiessu Jan 23 '14 at 16:03
  • $\begingroup$ You might want to elaborate on this “quadratic law” aspect. How exactly do you define that? $\endgroup$ – MvG Jan 23 '14 at 16:14
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    $\begingroup$ When you say "When a small sphere is at the edge of the large sphere its radius is $r$" that mean that if the small sphere is tangent to the large sphere (hence its center is at radius $R-r$) its radius is $r$? Then is the radius of each small sphere $r=\frac {\rho^2}{R-r}$ where $\rho$ is the radius of its center? $\endgroup$ – Ross Millikan Jan 23 '14 at 16:20
  • $\begingroup$ @RossMillikan I included an edit to the question. Does that clarify your question? Thanks. $\endgroup$ – triomphe Jan 23 '14 at 16:35
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    $\begingroup$ Yes, that clarifies it. With your function, at $x=0$ the radius is still $\frac r{(R-r+1)^2}$ which is not so small. You can probably do better by putting the second layer of spheres in the hollows between the ones on the first layer. These problems are very hard because irregular packings often are optimal. I was hoping the small spheres would go to zero radius and I could prove that you could pack infinitely many. $\endgroup$ – Ross Millikan Jan 23 '14 at 22:50

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