I read a story about somebody getting lost in a forest, and, apart from sympathy, the following mathematical problem came to my mind.
Suppose the lost person is sitting still at an unknown point in a square SxS forest (I saw a couple of questions on here where the person is assumed to be moving, but that seems harder).
A ranger starts search at the edge of the square, and they can spot the lost person within distance R.
What is the shortest-length path that the ranger can take to guarantee that they'll find the lost person? i.e. What is the shortest-length curve, whose extension by R covers the whole square. I wonder if it approaches one of the space filling curves (e.g. Peano) or is there a better way?