372 Frequently Asked Questions In Quantitative Finance
Urban planning
There are four towns positioned on the corners of a
square. The towns are to be joined by a system of roads
such that the total road length is minimized. What is the
shape of the road?
(Thanks to quantie.)
Solution
One is tempted to join the towns with a simple crossroad
shape but this is not optimal. Pythagoras and some basic
calculus will show you that the arrangement shown in the
figure is better, with the symmetrically placed crosspiece
in the middle of the ‘H’ shape having length 1− 1
√
3.
Obviously there are two such solutions.
Closer to the edge or the centre?
You have a square and a random variable that picks a
random point on the square with a uniform distribution.
