If you are given a list of cities and the times it takes to get between any two of them, what is the quickest way to visit every one of them? Solving this is a lot harder than it seems.
Stated mathematically, in a
complete weighted graph, what's the
tour of least cost?
This problem has been proven to be
NP-HARD. That means there's no shortcut to solving it, the only way you can do it is by thinking about all possible routes and picking the shortest one. A problem like "what is the shortest way to get from city A to city B, via another city", given the data above, is much easier.
This problem comes up everywhere - for example, in
computer science, if you have many programs to run in any order, all with different setup costs that vary depending on what program was previously running, you have to find the best route between them.
Or if you have a workshop and lots of different jobs to do, but each job takes a different set of tools - you have to decide what order to do the jobs, so you use the least amount of time getting the tools out and prepping the machines.