[mayoff]

The answer, with a diagram, can be found here:

http://mathworld.wolfram.com/SpiderandFlyProblem.html

[losvedir]

Ha, well that was a lot easier than I made it.

I ended up with the right answer by finding the minimum of the equation

    f(x,y) = 2(sqrt(1+x^2) + sqrt((6-x)^2 + y^2) + sqrt((15-y)^2 + 36))

which is the length of the path taken if the ant goes up to the ceiling at a point x to the East of straight up, and from there goes to the side wall at a point y to the south, and then goes from there to the center of that wall. (And then it's mirrored).

I should have thought to unravel the room! Errr.

[tzs]

To make the "minimize the equation" approach complete, other paths should be considered, such as instead of going up to the ceiling, going to the side and taking a side wall.

Some limits can be placed on the number of possible paths by considering some general properties of an optimal solution:

1. Assuming the start point is A, and the path leaves the start side at point B, the optimal path will take a straight line from A to B. Proof: if it were not straight, it could be replaced by a straight line which would shorten the path, contradicting the assumption the path was optimum.

2. Once the optimal path leaves a side, it will not return to that side. Proof. Suppose the path leaves the side at point B, then later enters at C, and then leaves again at D. The route from B to C could be replaced with a straight line from C to D, which would shorten the path, again contradicting the assumption that the path was optimum.

This cuts down the number of possible path templates greatly, since a given surface can only appear once in the optimal path, and only contain a single straight line segment.

I expect that a little more reasoning can deduce some more limits on possible optimal paths, to get it down to just two or three possible equations to maximize.

[d0mine]

for the unraveled room:

  sqrt(1+x^2) + sqrt((6-x)^2+y^2)+sqrt(12^2+z^2)+
  sqrt((30-z-y)^2+u^2)+sqrt((6-u)^2+1)

  40 at {x -> 0.75, y -> 7, z -> 16, u -> 5.25}

http://www.wolframalpha.com/input/?i=minimize%5B%7Bsqrt%281%...

your equation: http://www.wolframalpha.com/input/?i=minimize+2%28sqrt%281%2...