One morning at precisely 9:00 AM a monk begins walking up a mountain path. He takes his time, stopping several times to rest along the way. He arrives at the temple at the mountain's summit at precisely 5:00 PM that evening. The next day, the monk leaves the temple at precisely 9:00 AM and makes his way back down the path. Again, he takes his time and rests at several points along the journey. He arrives back at his original starting point at precisely 5:00 PM that evening. Is there any time when the monk is in exactly the same spot on both days? Click below to see the answer.

Since the monk isn't travelling at a constant rate of speed on his two trips, it's tempting to say that there's not necessarily a time when the monk is in the same spot at the same time on both days. However, such a time and place must exist. To see why, take a look at the following plot of the two trips.

Imagine that you can grab the lines on the plot and bend them however you like, you just can't move the endpoints, and the lines must stay within the bounds of the two axes. No matter how you stretch and bend the lines, they must cross somewhere.

To think of it another way, imagine there were two monks, one at the base of the mountain and one at the temple, and they started their journeys on the same day. If they were to begin and end their trips at the same time, they would have to pass each other on the path at some point during the day.

## 2 comments:

I like it!

Reminds me of the part of this about crumpling paper: https://www.math.hmc.edu/funfacts/ffiles/20002.7.shtml

I'm not sure sloshing your coffee really works. How do we know you can't slosh discontinuously?

Tim,

I'm going to have to read more about that theorem. It definitely sounds counter-intuitive, but a lot of the most interesting mathematical results do.

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