Question

2. One morning, exactly at sunrise, a Buddhist monk began to climb a tall mountain. A narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit. The monk ascended at varying rates of speed, stopping many times along the way to rest and eat dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting and meditation he began his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way. His average speed descending was, of course, greater than his average climbing speed. Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.

Answer 1

Here is not a rigorous proof, but it should suffice.

Since the path spirals, every point along the path corresponds to adifferent altitude, or height. Therefore, we can plot themonk's displacement as a function of time. We can place twomarks on our time axis: sunrise and sunset. We can also plottwo points on our height axis: base and summit. Then, thecurve will represent how the monk's height changes with time as hemoves up the mountain. At some points he will stop, so thecurve will be horizontal. We don't know exactly what thecurve will look like, but we know for certain that the curveconnects the two points (sunrise,base) and (sunset,summit). Seeexample graph below.

When the monk descends, we have a similar graph. His journeybegins at the point (sunrise,summit) but doesn't necessarily end atthe point (sunset,base). He will probably reach the basebefore sunrise. Nevertheless, any curve you can draw startingat the point (sunrise,summit) and ending at the point (time,base)will have to intersect the previous curve. Try it: there is no wayto avoid intersecting the ascent curve. Thus, this point ofintersection proves that they are at the same elevation at the sametime of day. See example graphs below.