In the Eternal Forest, the trees are perfectly circular, each having a diameter of exactly one meter. They are arranged in a flat, infinite rectangular grid. The center of each tree is ten meters away from the centers of each of its closest neighbors.

There are many paths through the Eternal Forest. Each path is infinite in length, constant in width, and perfectly straight. Trees don't grow on the paths, but every path will have tree trunks touching it on either side.

What is the narrowest possible path through the Eternal Forest?

I only started, but I looked only at paths with slopes 1/n (or n)

I came up with the formula for the width of the path

10sqrt(n^2-2n+2)*sin(arctan(1/(n-1)-arctan(1/n)) - 1

which yields the following

n     width
1     6.0711
2     3.4721
3     2.1623
4     1.4254
5     .96116
6     .64399
7     .41421
8     .24035
9     .104315260749
10  -.005

I wouldn't be suprised if a narrower path is possible along some other slope, but that's all I have time for now.

Posted by Jer on 2005-05-26 19:41:23