Points A and B are on a plane surface, 1 mile apart. Suppose you must walk in a path consisting of N straight lines from point A to point B, such that at all times your (Euclidean) distance to point B is decreasing. What is the longest possible route length (as a function of N)?
For N=2 it seems obvious the ideal path is to make AB the hypotenuse of an isosceles right triangle then walk the two legs. Total distance is sqrt(2).
For N=3 if we create a right triangle with AB a hypotenuse we can walk along the first leg and from there repeat the process for N=2.
From playing with Geometer's Sketchpad, the ideal first step seems to be to turn to an angle arctan(sqrt(2))=54.7 degrees and walk 1/sqrt(3) units. This brings you to a distance sqrt(6)/3 from B and repeating the process makes two more walks of length sqrt(3)/3.
The total distance is then sqrt(3).
How to prove this easily?
We see a recursive structure which I think generalizes. N=4 would be to find the ideal first step then apply N=3. Conjecture: first step angle arctan(sqrt(3))=60 degrees and distance 1/sqrt(4)=1/2. All four segments will be equal and total distance sqrt(4)=2.
Easily generalizes to any N.
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Posted by Jer
on 2019-01-15 10:01:19 |
N=3 and conjectured formula
