A couple of pictures, generated by a simple random-walk script. The colour indicates the progress of the location, going from dark to light. The direction for the 2 first pictures are not completely random (setting aside the random function’s limits). The point cannot go back to the last given point. So each time the point only has 3 choices for advancing.

Here I extended the code to draw 4 simultaneous walks, as well as having them draw semi random colours.