Hi Jim, This part: sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) sqrt((n-1)^2/n^2 + n(1/n)^2) sqrt(((n-1)^2+n)/n^2) sqrt(n^2-n+1)/n Should be: sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) sqrt((n-1)^2/n^2 + (n-1)(1/n)^2) sqrt(((n-1)^2+n-1)/n^2) sqrt((n^2-n)/n^2) sqrt(1-1/n) There are n terms total: One instance of (1-1/n)^2, and (n-1) instances of (0-1/n)^2. You were counting n instances of the latter, resulting in n+1 terms total. Tom James Propp writes:
Sorry if I'm being dense, but I don't see how Dan got
On Fri, Jul 6, 2018 at 4:57 PM, Dan Asimov <dasimov@earthlink.net> wrote:
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).
I get sqrt((1-1/n)^2 + (0-1/n)^2 + ... + (0-1/n)^2) = sqrt((n-1)^2/n^2 + n(1/n)^2) = sqrt((n-1)^2+n)/n^2) = sqrt(n^2-n+1)/n, not sqrt(1-1/n).
Am I suffering from end-of-workweek brain-fog?
Jim