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 $\frac{r}{R-r+1}$.
