Question:

There is a single path up a mountain. A hiker starts hiking up at 6 a.m. and reaches the top at 6 p.m. She stays at the top overnight. She starts down the mountain at 6 a.m. and arrives at the bottom at 6 p.m. On each day, the hiker travels at varying speeds. What are the chances that there was a specific spot on the mountain path that she encountered at exactly the same time of day on both the ascent and the descent?

.

.

C

A

P

T

A

I

N

I

N

T

E

R

V

I

E

W

.

.

Solution: With problems like this, it’s critical to listen to the statement of the puzzle and especially the challenge. “What are the chances. . . ?” Given the information, this is your clue that the chances are probably either 100 percent, 50 percent, or 0 percent. It’s unlikely that a puzzle like this requires a rigorous calculation of probability. More likely, it’s either possible or not, and since puzzles are unlikely to be posed if the situation is really impossible, you have eliminated 0 percent. The real problem, though, is explaining your answer. So the challenge is to think logically and re punctuate the problem. Here’s one candidate’s logical response:

The answer is 100 percent, and here’s how to visualize the proof. Imagine both the hiker’s trips taking place simultaneously. That is, the climber starts up at the same time her “twin” starts down. At some point along the way, regardless of whether one hiker stops to tie her shoelaces and the other one doesn’t stop at all, they will inevitably encounter each other as they cross paths. They must meet at some point along the path, and at that specific meeting place, they will fulfill the conditions of the puzzle.