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10/sqrt(97) - 1, or about .0153461651 meters. This path has a slope of 4/9.
Every path is parallel to a set of lines that go through the centers of trees. Paths with irrational slopes will eventually hit a tree.
A set of parallel lines with the slope p/q (where p and q share no factors) can be described with the equation px-qy=10c, where c is a different integer for each line. To determine the width of a path parallel to these lines, we must determine the distance between each line. This distance can be determined by finding the altitude of a certain right triangle much like the one drawn below.
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This right triangle is constructed by first drawing the line in the set that goes through a tree's center at the origin. This line has the equation px-qy=0. The next closest line has the equation px-qy=10. Using any tree on the latter line as the right angle, and the former line as the hypotenuse, we draw a right triangle with horizontal and vertical legs.
If the slope of the lines is p/q (in lowest terms), then the vertical leg of the triangle is 10/q and the horizontal leg is 10/p. The altitude is 10/sqrt(p² + q²), but the width of the associated path is one meter less than that, because of the radii of the trees.
Therefore, the width of any possible path through the Eternal Forest can be described as 10/sqrt(n) - 1 meters, where n is the sum of the squares of a pair of relatively prime numbers. |
