2 Ant Riddles

1. An ant is on the corner of the floor of a perfectly cubic empty room and wants to get to the farthest corner of the room, on the ceiling.  What is the shortest route?
2. Four ants are on each corner of a square, 2x2" Post-It note.  Each ant is facing the ant immediately to his left.  All four ants begin marching at once, each aiming for the ant immediately to his left.  Eventually the four ants spiral into the middle of the Post-It note, where they meet.  How far does each ant travel?

Solution:

1. There are several equally short routes.  One solution: walk to the middle of one the far edges of the floor, then walk up the wall from the middle of the floor up to the far corner on the ceiling.  One way to think about this is to knock down the walls so they fall to the outside.  The ant then walks in a straight line from its corner to the far corner.  
2. Each ant travels 2 inches - the length of one edge of the Post-It.  Even though they spiral inward, each ant travels as far as they would otherwise.  Since each ant's target ant is traveling in a direction perpendicular to his own path, the target ant is walking neither closer nor farther than it was to begin with.