| Intuitively the solution kind of makes sense. M must always be the highest point within the range. A and Z are at the same height and are the lowest point within the range. Imagine that the hikers are somehow connected (quantum entanglement, etc) such that one cannot physically move up or down if other cannot, and as one moves up and down the other must follow at the same height. Perhaps a better metaphor would be if the paths are cut into a wall, and you have inserted two pegs that are connected by a horizontal backing bar behind the wall. The bar may move up and down but will always remain horizontal. The pegs can slide left and right along the bar, and up and down along the paths, but must always remain horizontally level. Now, imagine that whenever a peg reaches a local max or min (peak or valley), a change in vertical direction may also cause a traversal along the opposite size of the peak/valley, thus allowing for forward progression. While one peg hits a vertical stop and makes horizontal progression, the other peg will simply move up and down along the same segment. This exercise is obviously not a mathematical proof, but does serve to make the proof feel a bit more intuitive. I'd love to construct such a "puzzle" myself and try it out on a bunch of different contours/tracks. |