Four Cities Solution

To solve this problem I take advantage of the symmetry of the Feynman solution and just analyze the left hand side, after doing so I just have to double my answer to get the total distance of road needed to connect the cities. My goal is to write an equation for the total length of the pattern and then take its derivative and set this equal to zero and solve for the roots. (minimization problem).

I introduce the variable x* to represent the horizontal part of the pattern.

Considering just half of the pattern, we only have to worry about this Total length = x + 2*L, (L is the length of the two slanted lines in the picture above) that was easy enough. Now I must express the length L in terms of the distance x, ( x is 1/2 x*), to do so I will construct a triangle where the hypotenuse of this triangle is the distance of the line.

Using the Pythagorean theorem, the length of the line is given by L= So now I can write the total lengh all in terms of x and take its derivitive. Total lengh= the derivitive of this is

Ok now, putting this all together, my total x* = 0.4226497308..., using L= to solve for L, L=0.5773502692.... Since there are four such L's that make up our pattern, I must multiply this result by 4 and add it to my x* distance to get a total distance of 2.732051099...

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