Question

Minimum-length roads. (from Problems for Mathematicians, Young and Old, Paul Halmos, MAA 1991)

Answer 1

Another way to look at it is what's the fewest number of lines that can connect all 4 dots?

Answer 2

6

Answer 3

Sorry, I've gone in the wrong direction (The answer to my question is 2, however)


(you can get to each of the houses this way, since the roads are connected, but it's also not the shortest path)

Answer 4

Aah, so the road "SYSTEM" must be the shortest, not the shortest path?

Answer 5

yeah, you have to minimize the total length of roads, so it's a calc problem (that I'm failing at setting up).

Answer 6

But it is a 1 mile by 1 mile square, wouldn't that just be a sqrt 2 diagonal?

Answer 7

yeah, but that's still not the shortest path - the roads need to intersect twice in the middle, via the hint.

Answer 8

I don't get the hint? Why 2 points?

Answer 9

I mispoke - there just exist two points in the middle, the roads don't intersect twice

Answer 10

Then each diagonal is
\[ \sqrt{(1/2)^2+y^2}\]
and the length of the center line is
\[2(1/2-y)\]

So total length is
\[L=4\sqrt{(1/2)^2+y^2}+2(1/2-y)\]
take derivative wrt y, and then set to zero to solve for y, then plug everything back in

Answer 11

\[ \frac{\mathrm{d}L}{\mathrm{d}y} = 4y \left( \frac{1}{4}+y^2\right)^{-1/2}-2=0\]

Answer 12

How did you get the length of the center line?