bug on the box problem A bug walks on the surface of a box ($L=B=1$, $H=2$), starting at a corner, $A$. You want to feed the bug but you also want the bug to walk the longest distance.  The bug takes the shortest path possible.  Where do you put the food to make it walk the longest?
The box floats in the air and the bug can walk on any face of box it wants.
Hint: its not $B$ Refer the image

If a curve gives the shortest path between two points then it is a geodesic. But any running line between two arbitrary points is  geodesic then the path length need not be the shortest. We can choose one among the many longer geodesics that are not unique on $\mathbb R^2$ including surfaces with discontinuous folds.
I assume we choose a corner $A$ and a farther off point from where to give the ant a run for food as start and target points but the logic holds for any point-pair, wlog. The shortest distance is along straight line whose path is has a directions inclined at
$$ \tan^{-1} \dfrac{2}{4}$$
to base square developed length $=4.$
There is a mathematical  definition of a minimum distance but none for maximum in unconstrained maximization.  Any arbitrarily longer path is a candidate when  inclined progressively less at
$$ \tan^{-1} \dfrac{2}{4n}$$
for arbitrarily large $n$.
The concept is from  Helix of a cylinder.
You can wind a thread between two points along helical geodesics in an infinite number of ways following progressively reducing pitch angles. 
Shown are paths on development of a square box 2 units high.

The modeling however in this way is constrained as stated to be along one or many of path minimizing geodesics.
