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?