Hillwalkers should love the Comte de Buffon (2020)
bencraven.org.uk
bencraven.org.uk
A straight needle crosses a gridline with probability (4 / pi) (l / t) where l is its length.
Surprisingly, a flexible noodle of length l also crosses an expected (4 / pi) (l / t) gridlines. To see why consider that the noodle can be approximated by a set of N straight segments of length (l / N). While the events of different segments crossing a gridline are correlated, the expected number of crossing for each segment is (4 / pi) (l / (N t)) and since, expectation is linear, the expect number of crossings for the noodle is still (4 / pi) (l / t).
* Practically useful
* Neat bit of approachable maths
* Surprising result
I've met hills I can ascend in 5 hours, and hills that took 9 hours, and both were the same gain in height.
In my country, backcountry travel is best described by time, not kilometres.
On a track, Igenerally estimate 1 hour for 300 meters ascent, 1 hour for 600 meters descent.
But in the bush, with no tracks but animal tracks, in the North Island or on the West Coast of the South Island, where the forest is denser, a good rule of thumb is 1 hour to ascend 100m, 1 hour to descend 200m.
In the eastern beech forests, which are typically more open, 1 hour for 200m up, 1 hour for 300m down is realistic.
The big issue of course, is that descending in untracked bush is a great way to find impassable waterfalls or bluff lines that require sidling or ascending to a ridge to bypass.
(Because pi/4 = 0.785 = ~0.8)