Charlie walks on vectors of (3,1) (-1,3) (-3,-1) and (1,-3). Another square he could walk, would have vectors of (5,2) (-2,5) (-5,-2) and (2,-5).
These vectors must only consist of four numbers: x, y, -x and -y. It can only be two numbers, and their negatives, so that all the sides of the square are equal in length. Each square must then contain four of eight vectors: (x,y) (-x,y) (-x,-y) (x,-y) (y,x) (-y,x) (-y,-x) and (y,-x). This can be in eight different orders. You begin at one point, then move either up, down or diagonally up or down. From each of these four points, you can then move left or right. From there, you must do the opposite of your first move, then the opposite of your second, to get back to your original position. The eight ways are:
(-y,x) (x,y) (y,-x) (-x,-y) (Up, right)
(-y,x) (-x,-y) (y,-x) (x,y) (Up, left)
(y,-x) (x,y) (-y,x) (-x,-y) (Down, right)
(y,-x) (-x,-y) (-y,x) (x,y) (Down, left)
(y,x) (x,-y) (-y,-x) (-x,y) (Diagonal up, right)
(y,x) (-x,y) (-y,-x) (x,-y) (Diagonal up, left)
(-y,-x) (x,-y) (x,y) (-x,y) (Diagonal down, right)
(-y,-x) (-x,y) (x,y) (x,-y) (Diagonal down, left)