The original problem was: given 4 cities arranged as the vertices of a square of side length 8 km find a way to connect the cities together using 22km of road or less. The solution can be seen as follows:
Constructing the road in this manner is "less" symmetric than just making the road with the diagonals of the square yet it is shorter. This intrigued me and I decided to see how much we could optimise this approach of leaving a space of length x in the middle. With a bit of simple calculus I found the length of the road is minimised for $$x = 2 \times (4 - \sqrt{\frac{16}{3}})$$ The length of the road is minimised to approximately 21.856 for this value of x.
Given all this I have numerous questions:
1. Could I have an intuitive explanation for why a "less" symmetric solution is actually better than the "optimally" symmetric solution of two diagonals even though the problem relates to a square which is very symmetrical.
2. Is this value of 21.856 the best we can do? Is there a better solution? Is there an optimal solution? If there is an optimal solution can we prove it is optimal?