Puzzle 1: Sometime on Monday, a climber ascends a mountain. Sometime on Tuesday, the climber descends from the mountain. Suppose the climber uses the same path going up as going down. Show that there is at least one point on the mountain where the climber spent the exact same time of day going up and going down.
Puzzle 2: Two mountain climbers stand at opposite sides of a set of mountains. By chance, they are both at the exact same elevation above sea level. They are very sympathetic, and decide that they always want to both be at the same elevation. Can the two climbers synchronize their actions in such a way that they are always at the same elevation, while managing to climb over the mountain ranges, and thus exchanging positions?
For simplicity, assume that the climbers are dots, that the mountain range is two-dimensional, and can be represented by a continuous function y=f(x) over some finite range [xa,xb], with f(xa)=f(xb), and that the climbers begin at the points xa and xb.
The answer to this puzzle is NO. Can you demonstrate this with a simple example function y=f(x), and a short explanation?
Puzzle 3: "Repair" the previous problem by making a simple assumption. Now show why the problem can be solved.
Puzzle 4: Show that there are always two places on the Equator that are directly opposite from each other (that is, the line connecting them passes through the center of the Earth), and that have the same temperature.
Puzzle 5: It is a common claim that hot water freezes faster than cold. Show that if this is really true, then under reasonable assumptions, this means something strange is going on! (My high school chemistry teacher reported that "It worked like a lead pipe cinch!" which actually frightened me a bit.)
I give up, show me the solution.