Conundrum of the men on the mountain.

[Vimto]

Another conundrum I can't recall the answer to was about a man who set off at the bottom of a mountain at 6am one day and walked up the only path to the top of the mountain which he reached at 6pm that day. He spent the night at the top and the following morning he set off again at 6am to walk down the mountain. The question is now asked whether he was bound to have been at the same place, at the same time on the two days? The answer was that he was as he must have passed himslf ,so to speak, when he somewhere on the path and it would have had to be at the same time as it couldn't have been at a different time as they were "together". Mr Always then gave a mathematical proof in additition to the logical one and it is this proof I am asking for. Can anyone please help?

[Rollo]

This is my "logical" proof.

The man starts his way down at the same time of day as he starts his way up.

Now let's suppose there are two men.

They both start their own way at the same time, but one starts from the top of the mountain
while the other starts from the bottom.


Since they are both using the same path they will definitely
meet at some point along the path and at some certain time of day.


This proves that the way up and the way down, when both are started at some certain time of day,
both have a point along the path which the two men pass at precisely the same time of day, irrespective of the speed of the journey at any particular time.


I can't provide the mathematical proof.

[Rollo]

If you draw two graphs on the same diagram (time on the X axis against distance on the Y axis) for the two journeys then there must be one, and only one, point where the two lines cross.

This point represents the same time of day and the same distance travelled for both journeys.

[Vimto]

Hello Rollo, yes, you are quite right; a little like the three day "kitco" gold charts where price is plotted against hours of the day and the plot line crosses indicating that the price was the same on at least two days at exactly the same time. Your logical answer was what I was trying to get at in my original question but you put it more clearly and I believe that "elegantly" is the word that mathematicians use. Thank you and well done.