**Please comment your solutions, questions and remarks.**.

The posts have been a bit advanced lately. Let us lighten the atmosphere by this riddle which admits a simple solution, though mathematicians tend to use calculus in solving it:

An Indian monk lives near a mountain. He wakes up in the morning at 6 a.m. and goes up the mountain. The journey takes six hours and at noon he is on the top. Then he meditates 18 hours and when it is 6 a.m. again, he goes down following exactly the same route as on the way up and it also takes six hours. Show that there is a point on the route which is passed by the monk at exactly the same time on both days.

1/5

This image is licensed under the __Creative Commons Attribution-ShareAlike 2.5 License__. I took from Wikipedia.

Is it just the start and end point? Clearly, these are the same on both days (f(0)=0, f(6 hours) = 1 trip), and if they are excluded from the possible points on the route than we can consider the case:

On the trip 1 he walks faster than trip 2 for almost the whole way, then at some time ‘t hours’ on trip 2 he increases his speed to make it there in 6 hours. In this case he would only be passing the same point at the same time if we included f(0) and f(6).

Right? I think we can construct cases where he only is passing the same point at the same time at the start and end points.

Oops sorry, misunderstood the question. Ignore that. I was imagining the monk going the same direction each day, but then I reread it.