Fun Fact: Divide the $n-$dimensional cube in half in each of $n$ dimensions, to create $2$ nsmaller cubes of edge length $\frac{1}{2}.$

Inscribe a ball in each of these subcubes, and then construct the smallest ball tangent to each of those $($and centered at the center of the original cube$)$ like so:

What happens to the diameter of the central ball as n gets large?

Surprisingly, the central red ball will stick out of encompassing large n$-$dimensional cube starting from a certain $n.$

Question: what is the smallest n when the red ball will stick out?