On Sat, Dec 27, 2008 at 2:49 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Suggestion: First guess the answer before doing any calculation.
PUZZLE: Let S and T be unit segments in 3-space with S parallel to the z-axis, with midpoint at (-x,0,0) and T parallel to the y-axis, with midpoint at ( x,0,0).
Connect each endpoint of S to one endpoint of T so as to get disjoint segments U and V.
Set m(x) := the (least) distance between U and V.
QUESTION: What is the limit of m(x) as x -> oo ?
On Sat, Dec 27, 2008 at 9:16 PM, Dan Asimov <dasimov@earthlink.net> wrote: (in response to a solution to the problem)
er, calculation.
Well, here's how I approached this problem without doing anything that felt like "calculation" to me. To visualize the problem, I held my hands far apart, with my left forefinger 2 inches above my left thumb, and my right forefinger 2 inches further away from me than my right thumb. The question is "how close does the line joining my forefingers come to the line joining my thumbs?" To exploit the symmetry of the problem, I rotated both hands towards myself by 45 degrees, which doesn't change the answer to the problem. Now the two forefingers are in one vertical plane, while the two thumbs are in another. Visualizing the two lines, it seemed immediately clear that the points of closest approach of the two lines were the points directly in front of me, since that's where the lines are in the same plane in the forward-back dimension. So the answer is the distance between the two vertical planes, which is 1/(root 2). Did I make this guess without calculation, as requested? Or was I cheating by solving the problem using a digital computer? -- Andy.Latto@pobox.com