A spider in one edge of a cube (length of all side = $l$) wants to get to an insect on the other edge of the cube. obviously the spider cannot fly and must walk on the sides of the cube to get to the insects. Find the shortest path possible. (See the image below)
I know the differential approach to solve this question (optimization problem) and the path is drawn on the above picture. But I want a geometric proof without using derivative. Just using geometric theorems (like pythagoras) and using simple algebra.
HINT
Let consider the unfolded cube on the plane.