A spider and a fly are sitting on a cube. The fly wants to maximize the shortest path to the spider along the surface of the cube. Is it necessarily best for the fly to be at the point opposite to the spider? ("Opposite" means "symmetric with respect to the center of the cube".)

Answer: no.
Suppose that the side of the cube is 1 and the spider sits at the middle of one of the edges. Then the shortest path to the middle of the opposite edge has length 2 . However, if the fly goes to a point on this edge at distance $s$ from the middle, then the length of the shortest path is

$$
\min \left(\sqrt{4+s^{2}}, \sqrt{\frac{9}{4}+\left(\frac{3}{2}-s\right)^{2}}\right) .
$$

If $0<s<(3-\sqrt{7}) / 2$ then this expression is greater than 2 .