A mountaineer starts rapidly climbing up a mountain trail at 6 A.M. He makes frequent and irregular stops to rest or eat. He reaches the summit at 6 P.M. At 6 A.M. the next day, he starts his way back following the same route, stopping only once to eat. He reaches the starting point at 6 P.M. Is there a point on the way where he passes at exactly the same time on both days?

Gnomon

Yes. If you plot a graph of height versus time for the ascent, you will a wiggly curve which always climbs, although it will not be straight. Now plot on top of it the same graph for the next day. You will have a wiggly curve which always descends. These curves must intersect at a point. At this point, the height it the same and the time is the same. Nov 08 02, 2:20 PM

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