You must cross a wide field, which is paved with circular flagstones, travelling only between centers of the stones. How far must you walk?

Question

The circular flagstones are arranged tangent to each other in the closest possible packing, i.e. with six stones surrounding each stone. The field width is W. Neglect the perturbations caused by the first and last stones you cross. What is the minimum, and maximum, possible values for your shortest possible path across? You may end at a point lateral to your starting point.

Answer 1

This question requires possibly drawing at least 3 (but more likely 4) circles to ascertain the geometry of the stones, and hence your possible path. Now, you may be lucky, and have a straight route across the field (length = W, the minimum possible), but at worst, you will have to angle at 30 degrees relative to a straight path across. That has a "lengthening factor" of 2/√3, for a total length of 2√3W/3. Your shortest route can't be longer than that, because the angle between two the centers of two adjacent circles in hexagonal symmetry is only 60 degrees, and that splits into 2 pieces with the smaller 30 degrees at most.

Now, if you want to up the ante a notch, so to speak, calculate the average distance, for a random orientation of the hexagonal grid of flagstones, relative to the edge of the field. You'll need a little integral calculus for that one, since you are integrating sec(θ)dθ from 0 to 30 degrees. Something like ln |(sec(θ) + tan(θ)| , if I recall. I leave it you to check the form, and solve the math bit.

-- Cheers, -- Mr. d.