Saturday Puzzle #24 – This Puzzle Has Its Ups and Downs

One of the nice things about living in Seattle is that on clear days we get a great view of Mt. Rainier. Considered an active volcano, Mt. Rainier is the third-highest mountain in the lower 48 states at 14,411 feet, and the most ice-covered, with 25 major glaciers covering 34 square miles (source).

Let’s imagine that at noon one day you set out from the base of Mt. Rainier and you reach the summit exactly 24 hours later, at noon the following day. You pause for a few moments to take in the view and celebrate your accomplishment and then you turn around and head back down the mountain. You may or may not travel the same route down the mountain but assume the descent takes exactly the same amount of time as the ascent: 24 hours on the dot. Thus, you spend precisely two days on this venture, one day going up and one day coming back down (in reality, the climb and descent would take less than a full day but I’m taking a bit of “puzzle license” here).

Here’s the big question: during those two days spent going up and down the mountain, was there a point where you were situated at the exact same elevation, at the exact same time of day? (For example, at 10:42:29pm on both days you were exactly 6,531 feet above sea level). Leave me a comment with your guess below.

Solution: Congrats to Al Pessot, Katy Gustafson, Dylan Gustafson, Mehmet Said, Doug Needham and Muzaffer Peynirci, all of whom found that there is sure to be one time of day when you are at precisely the same altitude.

The rigorous but esoteric way to prove this fact is to use something called the intermediate value theorem from calculus. But there’s a much simpler and more intuitive way to understand this puzzle. Instead of thinking about one person climbing and then descending a mountain, imagine two people – one starting at the bottom and one starting at the top, ascending and descending in parallel. At some point, those two climbers are guaranteed to pass each other, altitude wise, at precisely the same time.

Al Pessot came up with a formulation which makes this point more dramatically: imagine the two climbers are constrained to follow the same path up and down the mountain. At some point, they will literally bump into each other, and that point will be, of course, at the same altitude and time of day.

Finally, Katy Gustafson came up with an interesting “border case” solution I hadn’t considered: the ascent and descent start and end, respectively, at the same altitude and time of day (base camp at noon), which is absolutely correct. Sometimes the easiest solution is the one right in front of your nose. :)

2 thoughts on “Saturday Puzzle #24 – This Puzzle Has Its Ups and Downs”

  1. The answer is yes.
    Imagine that there are 2 of you and for simplicity assume there is one trail.
    One of you starts at the bottom and the other at the top.
    At some point during the day, the up hiker and the down hiker will meet and at that
    point both are at the same elevation.
    The same is true if a single person takes 2 days as in the puzzle.

  2. If you leave the base at noon and spend “precisely two days” on the mountain, you will arrive at the base at noon, where you will be situated at the exact same elevation and time of day as you were when & where you started.

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