Two curious problems:
1) Dudeney's Spider and Fly Problem (posed in an English newspaper in 1903): "In a rectangular room (a cuboid) with dimensions 30×12×12, a spider is located in the middle of one 12×12 wall one unit away from the ceiling. A fly is in the middle of the opposite wall one unit away from the floor. If the fly remains stationary, what is the shortest total distance the spider must crawl along the walls, ceiling, and floor in order to capture the fly?" Solution at Wolfram
2) Donald Knuth's Ant Problems: "With a cuboid 1x1x2, what is the furthest point on the surface from a corner, and which are the pairs of points with the greatest surface distance between them?"
Kotani's Ant Problem by Dick Hess solves both of these problems. Also described in Kotani's Ant Problem (book chapter) in Puzzler's Tribute: A Feast for the Mind edited by David Wolfe and Tom Rodgers.
Surface Distances on a Cuboid by Henry Bottomley discusses solutions to the above problems, along with many others.