IBM Ponder This #076 · August 2004

Puzzle for August 2004.

This month's puzzle comes from Joe Fendel, who writes:

Here's a puzzle I remember coming up with in high school. It was inspired by a request made one day that I come home immediately to do my chores, whereupon I would start walking home at a constant pace (I wasn't one to disobey), but I would try to find the longest possible route home (so as to put off doing my chores). Of course, I couldn't really walk in the opposite direction and claim to be "walking home." So I thought of the following problem:

Part 1:
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)?

Part 2:
Suppose the house is due north of the starting point, and on the way home, I always turn left (by less than 180 degrees). I still take a longest path according to the rules of Part 1. On the last leg of this longest path, I'm walking both northward and eastward (mostly eastward). What's the smallest possible value of N?

And of course your puzzlemaster has to add a "Part 3":
Suppose that in the setup of part 1, we have N=3; point A is 1 mile due east of point B; and the path is subject to the additional constraint that the last (third) line must be directed due south. Now what is the longest possible route?

Correct answers to all parts must be given.

Solution

Best opened after a real attempt

To be added.