Here's my solution to the "n-dimensional geometry puzzle" below: The center of the smallest sphere containing (1,0,...,0), ..., (0,...,0,1) in R^n must be at their centroid, which is C = (1/n, ..., 1/n). The distance from the point C to any of the basis vectors is sqrt(1-1/n). The distance from C to the origin is sqrt(1/n). Since sqrt(1-1/n) >= sqrt(1/n) for all n >= 2, (exercise), sqrt(1-1/n) solves the problem. As far as I know, only Tom Karzes sent in a solution (almost instantaneously). —Dan ----- Puzzle: ------- In n-dimensional space R^n, find the radius R = R(n) of the smallest sphere containing (whether inside or on the surface) the standard basis vectors {e_k} = {(1,0,...,0), ..., (0,...,0,1)} and the origin 0 = (0,...,0). I.e., R(n) = inf {r > 0 | for some c in R^n ||p - c|| <= r for p = 0 and all p = e_k} Apologies if this has been asked already; I don't recall. -----