Pensieve header: The 1206 riddle for MAT 257.

Consider the (2^n) yellow balls of radius 1 with centers at the (2^n) vertices of the n-dimensional cube (({-1,1})^n). Let (Subscript[C, n]) be the smallest box containing these balls, and let (Subscript[B, n]) be the largest blue ball centered at 0 bound by these balls. Compute (lim_(n->∞))((Vol((Subscript[B, n])))/(Vol((Subscript[C, n])))).	PS. I wouldn’t be asking, if I didn’t think the answer was worth knowing.

GraphicsGrid[{Rasterize/@{
Graphics[{Yellow,Disk /@ Tuples[{1,-1},2],Blue, Disk[{0,0},(Sqrt[2])-1]}, Frame->True],
Graphics3D[{Yellow,Ball /@ Tuples[{1,-1},3],Blue, Ball[{0,0,0},(Sqrt[3])-1]}, ViewPoint->{2.06766,-2.67826,-0.0415505},ViewVertical->{0.465871,-0.399345,0.789613}]
}}]