Question

A monk set out from a monastery in the valley at dawn. He walked all day up a winding path, stopping for lunch and taking a nap along the way. At dusk, he arrived at a temple on the mountaintop. The next day the monk made the return walk to the valley, leaving the temple at dawn, walking the same path for the entire day, and arriving at the monastery in the evening. Must there be one point along the path that the monk occupied at the same time of day on both the ascent and descent? (Hint: The question can be answered without the Intermediate Value Theorem.) (Source: Arthur Koestler, The Act of Creation.)

Short answer

Question: Explain, using a thought experiment, how we can prove that there must be at least one point along the monk's path where he was at the same time for both ascent and descent journeys. Answer: By visualizing another monk (Monk B) starting from the temple at the same time as the original monk (Monk A) and taking the same path but in reverse, we can acknowledge that both monks will pass each other at some point during their journeys. At this point of intersection, both monks have traveled the same time distance in their ascent and descent. Thus, there must be at least one point along the path where Monk A was at the same time of day for both ascent and descent journeys.

Step-by-step solution

The problem describes a monk who is walking from a monastery in the valley to a temple on a mountain top (called ascent) and back again (called descent). We should imagine the entire path of ascent and descent as mapped by time taken. They can be thought of as two separate time-dependent paths, which we are trying to find their intersection. # Step 2: Create a "mirror" experiment #

Now, imagine that there is another monk (which we will call Monk B) who starts at the same time from the temple on the mountaintop, taking the same path down to the valley and then back up again. In this scenario, Monk B will follow the exact same time-based path of ascent and descent as the original monk, but this time in reverse order. # Step 3: Identify the point of intersection #

Since both Monk A (original monk) and Monk B (imaginary monk) start at the exact same time and on their way, they will pass each other at some point during their journeys. Note that at this point of intersection, both monks have traveled the same time distance, experiencing the same amount of time journey passed in their ascent and descent. # Step 4: Conclusion #

There must be a point along the path where the actual monk encountered with Monk B (imaginary), which mean the actual monk was at the same point in his ascent path and descent path at the same time of day. This proves that there is at least one such point along the path that the monk occupied at the same time of day on both the ascent and descent journeys.

Key concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that provides a guarantee about the presence of certain values within a continuous function over a closed interval. Simply put, if you have a continuous function that produces different outputs (values) at two points, the IVT tells you that the function will take on every value between those two outputs somewhere within that interval.

For instance, if you're on a road trip and start at sea level and then drive over a mountain pass, you have to pass through every elevation between the start and the peak. IVT mathematically affirms this, as long as the elevation changes continuously without jumps (which it naturally does). In the monk's journey, utilizing IVT can lead to understanding that, because the monk's ascent and descent are both continuous processes, there must be at least one elevation (or point on the path) that the monk crosses at the same time of day on both trips, as long as the monk's journey from and to the valley occurs within a single continuous day. The theorem is often used implicitly to rationalize why certain outcomes must happen, even though the exact point may not be readily identifiable without further analysis.

Mathematical Proof

A mathematical proof is a logical argument that establishes the truth of a given statement beyond any doubt. It's a process of using existing known truths, definitions, and axioms to show that a new statement must also be true.

In the context of our monk's journey, a mathematical proof doesn't necessarily involve complex calculations or the direct application of theorems like the IVT. Instead, we've constructed a 'mirror experiment' with an imaginary second monk, and through logical deduction, we demonstrate the existence of a point in time and space where the monk's paths intersect. This kind of intuitive and conceptual proof showcases the beauty and elegance of mathematical reasoning, where sometimes the solution requires thinking outside the traditional procedural box. With the clever idea of considering two monks on the same path but starting from opposite ends, we can conclusively say that there is definitely at least one point they share at the same time without specifically finding that point.

Conceptual Analysis

Conceptual analysis is about thoroughly understanding and assessing the underlying principles and ideas of a problem. It goes beyond just applying formulas or theorems mechanically – it’s about figuring out what the problem is fundamentally about and then using that deep understanding to find a solution.

In the monk problem, instead of directly employing the Intermediate Value Theorem, we use conceptual analysis to reframe the scenario with two monks. This exercise not only enriches the understanding of the problem but also presents a different viewpoint to arrive at the conclusion. Conceptual analysis can often reveal new insights, leading to a more intuitive grasp of the situation. It's a critical thinking skill that involves breaking down complex situations into basic elements and is an essential part of problem-solving in mathematics and many other fields. Hence, always consider different perspectives or setups of a problem to access more straightforward solutions, much like the imaginative second monk pathway that helps us understand the result beyond just the mechanics of the IVT.