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Impossible vs. Insufficiently Clever

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I find people will often say something is impossible, when really they just aren't s mart enough to figure it out.

Physicists (and the Discovery channel) love pointing this out about time-travel: No law of physics, except for perhaps the law about increasing entropy, actually says that traveling backwards in time is impossible.

My favorite example of this: what's the minimum length of wire needed to connect the 4 corners of a unit square (square with edge=1, figure A below) to each other?   (The same section of wire may be involved in multiple connections, as demonstrated in figure B).

You may say, "That's obvious, the answer is 'C'". But can you prove it? Afterall, if you can't prove it, how can you really be so sure it's true? When I ask people, their proofs usually involve something like "it's so obvious, it can't possibly be any shorter". (If you know the answer, ask a smart friend who doesn't and you can see what I'm talking about)
It is interesting to see people passionately prove a wrong answer.

The flip side of this is when a problem is indeed provably-impossible, yet somebody insists on trying to find a solution. There are lots of math examples here too. And lots of Dilbert examples too.

The middle ground: And then there's a whole middle range. Perhaps you can't figure out a solution, but you can show that whatever it is, it must have some certain properties.

Comments

  • Anonymous
    February 06, 2006
    Lookup steiner points.  The shortest distance is, according to the link below, 2.732.

    http://www.mathreference.com/gph,stein.html

    "Let's connect the corners of a unit square.  Without steiner points, the spanning tree has length 3, with 3 of the 4 sides drawn in.  Your next impulse is to connect the four corners to the center, introducing one steiner point.  This gives an edge length of 2.828, a definite improvement over 3.  Next, split the steiner point in two and pull the two points apart, towards the left and right sides of the square, until the angles are 120°.  This gives an edge length of 2.732.  This is the best steiner tree for the square."
  • Anonymous
    February 06, 2006
    The comment has been removed
  • Anonymous
    February 06, 2006
    Clearly you need "the theory".
    You assumed (and didn't prove) that "H" shape is the best.

  • Anonymous
    February 06, 2006
    Isn't this the kind of shape you get if you play with soap in the bathroom? :-)
  • Anonymous
    February 06, 2006
    The comment has been removed
  • Anonymous
    February 06, 2006
    The comment has been removed
  • Anonymous
    February 06, 2006
    The comment has been removed
  • Anonymous
    February 06, 2006
    The comment has been removed
  • Anonymous
    February 07, 2006
    The real answer is 0.

    You're missing the obvious solution, like the Microsoft solution in coming out with C# on .NET to replace C++ -- just mash that square into on little blob and you don't need any darn wire.
  • Anonymous
    February 10, 2006
    Sometimes we think we can make a decision, but it turns out the decision is already made for us by the...
  • Anonymous
    June 12, 2007
    Here's a little number puzzle quiz. Fill in the digits: ABC + DEF GHI Where each letter represents a