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
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Puzzle:
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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.
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