This week's "riddler classic" on $538$ is quite the puzzler (as the riddler classic always is). However, I can usually deduce solutions by researching relevant material or just using what's already up in the 'ol noggin. The question states:
Consider four towns arranged to form the corners of a square, where each side is $10$ miles long. You own a road-building company. The state has offered you $\$28$ million to construct a road system linking all four towns in some way, and it costs you $\$1$ million to build one mile of road. Can you turn a profit if you take the job?
Extra credit: How does your business calculus change if there were five towns arranged as a pentagon? Six as a hexagon? Etc.?
I don't need a solution to the problem (as that's the fun!), just want a jumping off point. Simply was wondering how to classify the problem. I've been trying to find similar problems and whatnot to help me along, but I'm struggling. I considered graph theory right off the bat, but there's no nodes to work with! I've also thought of using Lagrange Multipliers, but I'm not quite sure that's applicable here. Lastly, I even considered using mathematica to basically drop particles on a polygon then apply graph theory to that, but that seems rather over the top.
Any hints would be greatly appreciated.
My at-a-glance guess: Steiner minimal trees.
See, e.g., Gilbert and Pollak's (1968) Steiner Minimal Trees (pdf) or Courant and Robbins' (1941) "What is mathematics?" as mentioned in the blog post here.
Side-note: Gilbert-Pollak's $\sqrt{3}/2$ "Steiner ratio conjecture" [at the end of p. 23 above] was believed to have been proved successfully; however, an error was later discovered.
Cf. MO 144081 for more details.