Pardon me, but here is another problem which surfaces from my own extension on a solved problem, i.e. to say there is no "answer key" to refer to, for the problem I am proposing. I tend to extend my own thoughts and create problems which become puzzle for myself. So here it goes.
There is a famous problem on circles which asks for the maximum possible radius of 3 equal circles, which can be placed inside a circle of defined size. The required placement of circles is as shown below.
By applying certain basic geometry principles, it can be easily obtained that the ratio of the radius of each of the three smaller circles to the larger one which contains them is (2√3 – 3) : 1
Now I was just wondering, that analogous to this, what could be the largest possible radius of each of three equal spheres, that can be kept inside a sphere of defined size. Let us take it of unit radius for purpose of discussion.
I am trying my best, but am failing to imagine the 3D analogous figure here, and what would be the ratio of radius of the three smaller spheres to the larger one.