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12 June 2008

Simple puzzles and engineers

Recently, I've stumbled upon a mind puzzle suggested on etd's blog. It seemed pretty simple, so I gave it a little thought...

Just for the sake of completeness, here is the puzzle:

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.
Here is my go at it:
  • Considering that the path is an ascending spiral [from bottom to top of the mountain], it is possible to graph its height vs length as an injective function. This means that any point of height is occupied by a spot in the path, representing a single determined distance point from one of the ends of the path;
  • The monk will have to go through every point of height (or distance) to reach the top. He won't jump. This also means that height or distance can also be graphed against time as an injective and continuous function. At a given point of time, the monk cannot occupy two different height or distance spots;
  • The following sketch is my representation of the monk's distance covered along the day (distance vs time). I've superposed the graphs for the climb and descent.
  • Knowing that both climb and descend started at the same time, no matter how crazy the curves are (representing rests and different instant speeds along different bits of the path) they *have* to meet somewhere because they are continuous. The meeting place of the lines represents a single spot where the monk was at exactly the same time on both journeys.

After having conceived this analytical solution to the problem, I've realised this other one:
  • Both journeys are made on different days. So an equivalent situation would be to have two monks at the same time starting their journeys in opposite directions - one climbing and the other descending. They will meet along the way.
And this was just a quick (but rare) peek inside the mind of an engineer.

Cheers, PJ.

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