The monk

Once upon a time there was a monk. He lived at the bottom of a not-so-high mountain in Tibet. He had a peculiar habit. The last day of every month, precisely at sunrise, he left his hut at the bottom of the mountain and walked up a path to the top, timing it so that he arrived at the top precisely at sunset. He meditated all night. The next morning—the first day of the new month—precisely at sunrise, he left the top of the mountain and walked down the same path, timing it to arrive back at his hut precisely at sunset.

Was there necessarily a point on the path at which he arrived at the same time of day on each of the two days—going up and coming down?

Approaches

Intuitively, there's no such point.

Intuitively, there is such a point.

Convince others.

Confusion about times of sunrise and sunset changing from one day to next.

Confusion about times of sunrise and sunset changing between top and bottom of mountain.

Assume constant speed.

Does that help?

Try making timeline.

Does the failure mean that there's always such a point, or just that you haven't found a way to avoid it?

Graphs of height vs time of day.

Will they always have to cross? Why?

"They cross at the same point, but not the same time."

At what different times do they cross?

Confusion between shape of graph and mountain.

Imagine two monks, one going up and one coming down the same day, mimicking the actions of the original monk. They must meet.

Is the two-monk model a valid representation of the situation?

If the endpoints aren't on the path, there's not necessarily a point.

Are the endpoints on the path?

Simulation

Get into the head of a teacher teaching a class session with this problem.