You have $n+1$ points in $n$-dimensional Euclidean space, such that the mutual distance between each pair of points is the same. Now imagine an $n$-dimensional generalization of a sphere, such that all of these $n+1$ points lie on this sphere. Let's call this circum-hypersphere. Similarly, we can define an in-hypersphere, such that the midpoints of each $n$-point subset lie on the in-hypersphere.
How man dimensions $n$ are needed to make the $n$-dimensional hypervolume of the circum-hypersphere more than a billion times larger than the one of the in-hypersphere?
Probably this way of asking the question is a bit abstract, but here are some examples in low-dimensional spaces to explain what I mean:
- In $n = 2$ dimensions we have $n+1=3$ points. If we arrange them such that the mutual distance between each pair is the same, it results in an equilateral triangle. The "circum-hypersphere" is also called circumscribed circle is this case and the "hypersphere that contains all midpoints of $n$-point subsets" is the incribed circle, as it contains all midpoints of the triangle edges.
- In $n = 3$ dimensions the $n+1=4$ points form a regular tetrahedron. The circumsphere is again trivial and the insphere is constructed such that it touches the midpoints of all the triangular faces.
This problem seems to be hard in terms of the length of the required calculations, but there is an easy way. Hence, the answer with the simplest explanation will be accepted.