jmc

In the above diagram, the green dot is the gecko and the purple dot is the fly. We can ``unfold'' the walls that the gecko traveled along, as below, to represent the gecko's path in two dimensions. This unfolding does not change the length of the gecko's path, so in order for the gecko's path to be minimal before unfolding, it must be minimal after unfolding. In other words, it must be a straight line after unfolding. Now, besides the side walls, the gecko can travel along the front, back, and ceiling. Suppose that among these, it only travels along the front wall. The walls the gecko walked along unfold as such: The gecko's path is the hypotenuse of a right triangle with legs 6 and 22, so its length is $\sqrt{6^2+22^2}=2\sqrt{3^2+11^2}=2\sqrt{130}$. By symmetry (the gecko and the fly are exactly opposite each other in the room), the path length is the same if the gecko only travels along the back wall and side walls.

Now suppose the gecko only travels along the ceiling and side walls. These walls unfolded become: The path is the hypotenuse of a right triangle with legs 8 and 20, so its length is $\sqrt{8^2+20^2}=2\sqrt{4^2+10^2}=2\sqrt{116}$. (We'll keep it in this form because it makes it easier to compare with the other cases.)

Finally, the gecko may cross both the ceiling and front wall (or back wall; the cases give the same results by symmetry). The unfolded walls then look like this: The path is the hypotenuse of a right triangle with legs 16 and 14, so its length is $\sqrt{16^2+14^2}=2\sqrt{8^2+7^2}=2\sqrt{113}$. Of the three cases, this is the smallest, so the answer is $\boxed{2\sqrt{113}}$.