Mountain climbing problem

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is a problem of finding the conditions that two functions
forming profiles of a two-dimensional mountain must satisfy,
so that two climbers can start on the bottom on the opposite sides of the mountain
and coordinate their movements to meet (possibly at the top)
while always staying at the same height.

This problem was named and posed in this form by James V. Whittaker (1966),
but its history goes back to Tatsuo Homma (1952), who solved a version of it.

The problem has been repeatedly rediscovered and solved independently in different context
by a number of people

In the past two decades the problem was shown to be connected to
the weak Frรฉchet distance of curves in the plane,[1]
various planar motion planning problems in computational geometry,[2]
the inscribed square problem,[3]
semigroup of polynomials,[4] etc.
The problem was popularized in the article by Goodman, Pach & Yap (1989),
which received the Mathematical Association of America's Lester R. Ford Award in 1990.[5]

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It is easy to coordinate the climbers' movement between the peaks and valleys
(local maxima and minima of the functions).

The difficulty is that to progress, the climbers must occasionally go down the mountain,
either one or the other, or both climbers.

Similarly, either one or the other climber must backtrack towards the beginning of the journey.

In fact, it has been observed that for a mountain with n peaks and valleys the number of turns
can be as large as quadratic in n.[1]
These complications make the problem unintuitive and sometimes rather difficult,
both in theory and in practice.