In the top example. the red line only represents half the circumference of the coin. In the bottom example, the red line is the entire circumference. Hence, the top line is half the length of the bottom.
Hmm. No, I believe @LDoBe had it right from the start. Itâs certainly possible that there could be a particular mountain path upon which a particular two-way journey over two days could result in a confluence as described in the problem. But consider a mountain path with a pair of large rocks near either end⌠a pair of rocks that happen to have a gentle slope on the ends that face the interior of the path, but a fairly steep and arduous approach on the ends that face the ends of the path. Shortly after beginning your ascent, you might spend an hour clambering over that first rockâs steep end, but then youâd easily lope down its far side and continue up the hill. As you neared the top, youâd stroll up the second rockâs gentle slope, and then bound down the steep side like a mountain goat, reaching the summit shortly thereafter. Your return trip would be similar, with a slow and arduous beginning, and an easy descent the rest of the time. Temporally speaking, your ascent would be half over before youâd ascended half the distance of the path, and your descent would similarly take a longer period of time for the first half than for the second half. So at no point in both days would you be in the same area of the path. Lunchtime would see you some distance short of the halfway point along the trail.
But on other trails, ones where your upwards and downwards paths have overlapped by lunchtime, yes, that point would exist.
Nope. Since both the route up and the route down overlaps spatially, and since the time it takes is constrained to only between sunrise and sunset (and we assume the two consecutive days are identical, perhaps the same day of the year one year apart), then there will always be a point during both trips where you would have been at the same time.
Imagine you and your clone doing it on the same day, you starting at the bottom, and your clone starting from the top. If you both take the same amount of time to complete the trip, regardless of varying speed, youâll still âcross each otherâs pathsâ at some point.
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Okey doke. In my example, it would just be after lunch. Thanks! Well-considered.
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Hey, come to think of it, as long as youâre both on the trail at 7:00 am, it doesnât matter if the ascent took three times as long as the descent⌠or indeed if you spent all day huffing up a 40% grade and then just slid down the whole thing on a sled in three minutes, as long as youâre done by the dusk deadline, youâll run into each other at some point. 
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ProfessorDumb this is very much true. the sound of a record degrades as it goes towards the center, a known issue for music producers.
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