TITLE: You are standing at the origin of an "infinite forest" holding an "infinite bb-gun"
QUESTION [25 upvotes]: I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes...
You are standing at the origin of an "infinite forest" holding an "infinite bb-gun." The "trees" in this forest are at the lattice points all around you. (The lattice points are like those on graph paper and they align with the cardinal directions: N, S, E, W.) The "forest" is Euclidean in the sense that the trees have no width. To hit a tree with your bb-gun you must aim perfectly at it. 
You would, for example, hit a tree if you fired the gun due north, south east or west. (Your bullets also have no width.)
A. You fire the gun in an arbitrary direction without bothering to aim. What happens?
B. You get a new bb-gun and the bullets have a little width to them. ($\delta$?) You fire the gun in an arbitrary direction without bothering to aim. What happens?
C. All of the trees are removed that have coordinates whose absolute values are not perfect squares. (So, only points such as $(25, 100)$ and $(4,-1600)$ remain.)  Again you use width-less bullets. You fire the gun in an arbitrary direction without bothering to aim. What happens?
D. Again, only with perfect squares, but now the bullets have width. What happens?

REPLY [3 votes]: The problem with trees of finite width at all lattice points is called "Polya's Orchard Problem".  Other variants such as higher dimensions and different shapes of the orchard are in the literature under the same name, or variations on visibility, lattice point, and orchard.  Sample search result:
http://www.rose-hulman.edu/mathjournal/archives/2006/vol7-n2/paper9/v7n2-9pd.pdf
For the problem with zero-width trees and visibility from the origin (ie., primitive lattice points) I think the $6/\pi^2$ density is attributed to Chebyshev in the 1800's, but it seems like the type of result that could have been known considerably earlier and frequently rediscovered.