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?

(In reply to All done! by Jer)

"4/9   .015346165134" appears to be  the smallest that you show and agrees with the value I posted below which I calculated from 10/sqrt(97) - 1. Notice that 97 = 4^2 + 9^2, and this is no accident!

In a forest with trees centered on a unit grid, the lines that have slope q/p, where q and p have GCD 1, and pass through the centers of at least two different trees, are spaced at 1/p vertically and 1/q horizontally. The distance between these lines is found to be 1/sqrt(p^2 + q^2). This must be multiplied by 10 to get the similar distance in Tristan's Eternal Forest.

Edited on May 27, 2005, 7:41 pm

Posted by Richard on 2005-05-27 19:31:05