You walk up a mountain starting at 6 AM and get to the top at 6 PM. You spend the night at the top and on the next day, you start walking from the top at 6 AM and get to the bottom at 6 PM along the same path. You do not necessarily travel at the same speed on the two days. Show that there must be at least one clock time of day when, while going up and coming down, you were at the same height.
Solution: Let d be the height of the mountain. Suppose you plot a graph of time versus height when you go up and when you descend. The graph will start at 6 AM on the time axis on the first day and reach a height of d at 6 PM. The shape of the graph will depend on the speed with which you traveled. On the next day, at 6 AM the return graph will start at a height d and reach a height zero at time 6 PM. Regardless of the shapes of the outbound and return trips, there must be at least one point between the top and bottom where the graphs cross. At this time, you were at the same height at the same clock time on both days.