A sphere is reconstituted into n identical smaller spheres, keeping the total volume constant.
Let a be the ratio of the total surface area of the smaller spheres to the surface area of the original sphere.
Let b be the ratio of the radius of the original sphere to the radius of each of the smaller spheres.
Let c be the ratio of the volume of the original sphere to the volume of each of the smaller spheres.
What is the minimum integer value of a+b+c?
For the large radius R and small radius r
c = n = R^3/r^3
So, b=n^(1/3) and a=n^(2/3)
and the question becomes:
For what integer n is n^(2/3) + n^(1/3) + n also integer?
Dismissing the trivial case n=1, this next happens at n=8,
where a+b+c is 14
Edited on November 25, 2019, 9:32 am