Impossible vs. Insufficiently Clever

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.

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1. Roger Wolff says:

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."

2. barrkel says:

I see Roger has gotten ahead of me, but I needed to work it out without the help of the theory.

I wrote an equation that described the total wire length when using a "squeezed H", with an adjustable "waist", length x:

x + 4 * sqrt((1 – 2*x + x*x)/4 + 1/4)

Using a test value for x of 0.5 (trying to neutralize the 1 with 2*0.5, yet knowing that x*x < x for 0 < x < 1. Using x as 0.5, the total wire length is:

2.7360679774997896964091736687313

3. dimkaz says:

Clearly you need "the theory".

You assumed (and didn’t prove) that "H" shape is the best.

4. David Srbecky says:

Isn’t this the kind of shape you get if you play with soap in the bathroom? 🙂

5. Roger – you’re a step ahead of me. 🙂

I didn’t want to come out and say "C" was the wrong answer because that would give things away. It’s very interesting to see somebody passionately "proove" that C is correct. Since C is not correct, their proof will obviously be wrong. So at what point does the lightbulb go on that "I can’t proove C is correct, so maybe C is incorrect and their is actually a better way".

I guess if I had thought ahead, I would have moderated the comments so that nobody would give away the answer. Alas, maybe next time 🙂

6. Roger Wolff says:

David,

Yeah, my high school geometry teacher did a pretty neat demo for us.  He brought in two parallel pieces of plastic, connected by metal rods at the vertices. He dipped the contraption into a bucket of soapy water, and voila! — steiner points.

7. SteveJS says:

Cool … Steiner points.  🙂

I remember a somewhat similar problem in a freshman college course on tricky Math problems.

Rather than connect all the points, you must instead prevent any line from intersecting 2 edges without intersecting at least one line segment interior to the square.  Obviously the solution to the problem above satisfies the condition.  But is there a better one that uses line segments of less total length?

8. barrkel says:

dimkaz: I see no proof in Roger’s entry. It talks about ones next "impulse", and then splits the point in two, and then asserts that "this is the best steiner tree for the square".

My only goal was to prove that "C" wasn’t the most optimal entry, and I did by providing an example that was better than it.

9. Been through too many "paradigms" says: