A famous puzzle is based around the seven bridges of the city of Königsberg in Prussia (now Kaliningrad, Russia) . The task is to devise a route for the good citizens on their daily perambulations of the town to cross all seven of the bridges once and once only.
From a walking enjoyment point of view this appears to be a useful commission. Many people like to be comprehensive in their explorations while avoiding retracing their tracks. The cautionary learning point is that the puzzle has no solution; Leonhard Euler proved this mathematically in 1736 and in so doing laid the foundations of graph theory and a whole new branch of mathematics!
Small complex and interesting places such as Drum Manor (or Königsberg) can lend themselves to the plotting of a more complex route to visit all the best bits, lengthening and enriching your walk (even if small retraces are required). You may not invent a new branch of mathematics but you will maybe solve a few human problems as you clear your head along the way!