Imagine an infinite square grid in the plane. Starting from a chosen point on the grid (shown in red in the figure below), we move according to certain rules governing elementary moves, represented here for convenience by vectors in the plane. The question is simple: which grid points can we reach by applying these rules as many times as we like?
Starting from the red point and using only the two moves shown, we can reach all the gray points.
Hitting the target every time
Exploring several sets of elementary moves that can be combined in any order—what we will call a positive integer linear combination—reveals that we can sometimes reach every point on the grid, but sometimes only certain points.
