Here is another of my favourite riddles that I think, in its own small way, is a tribute to logic and math and the way they fascinate those who often find themselves pondering over the nuance of 'riddle' that can be seen in everything around us. So I encourage you to play along :).
Samwise Gamgee has a square plot of land, each side being 1 unit. One day, Sam finds out that the dark Lord Sauron has a telephone line that he uses to speak with a traitor amongst the hobbits.
Gandalf informs him that the telephone line runs in a straight line parallel to the ground and passes beneath the square plot of land, but he does not know its location. Sam decides to dig up around the perimeter of his land to discover the telephone line, but Gandalf says it is not necessary to dig around the entire length of 4 units.
Sam brightens up, and says "I know what you mean. I can just dig 3 sides and still discover it. For even if the phone line runs along the fourth side, I will still detect it at the end points ! "
Gandalf shakes his head. "No, Sam. You are on the right track, but you can do better than that."
What solution does Gandalf have in mind for the optimum length of the "digging curve" ?
As with the other one, there are many different solutions. It's a problem of optimisation, so any given solution can be tested to see how it compares to the others. I think I know the optimum solution, but of course if someone finds one that is better, then I stand corrected.