mountain climbing problem

Question

The wikipedia article on the mountain climbing problem mentions two things which I am unable to reproduce on a number of test mountains I drew:

• that both of the climbers may occassionally need to go down the mountain
• that for a mountain with n peaks and valleys the number of turns can be as large as quadratic in n.

Perhaps it doesn't help that the article's illustration is a rather mundane mountain that exhibits none of these two properties. Apparently I am missing something. Can someone provide the drawing of a mountain that would satisfy either one or both of the above conditions? Also, why would a mountain with areas of constant height complicate matters?

Answer 1

For your first question, here's an example where climbers $A$ and $A'$ start at their respectively marked locations; and in order for both to reach the glorious icy peak at $P$, at some point $A$ must backtrack segment $DC$ while $A'$ simultaneously backtracks segment $F'E'$:

For both climbers, they are simultaneously going away from the peak of Mount $P$. I'm still trying to think of an example where they are also both decreasing their altitude as they move away from the peak; but perhaps this is sufficient for your needs.

Answer 2

Just to add to @Chas Brown's answer — Here's an SVG animation of the two climbers solving Chas's example. (I tried to upload this animation to Wikimedia Commons and thence to Wikipedia, but it turns out Wikipedia doesn't really support animated SVGs, so it's not particularly viewable there.)