I'm frequently asked how I manage to draw my mazes in such a way that I can guarantee that there is exactly one path between the entrance and exit. In my initial drawing phase of a maze, I follow a very simple rule: "no existing pathway can join another". Making only "outward branches" ensures one path between entrance and exit in the same manner that adding an node/edge to a binary tree doesn't make loops. However, I don't just start on one corner and grow my maze to other corner. I draw where my whim takes me and end up making hundreds of little disconnected sub-mazes.
the four color problem - though I don't need an optimal coloring, and since I do it by hand, I use as many colors as is convenient, usually between six and eight.
At this point, I prove that I've produced a maze. I do this with floodfill and the idea that a maze is isomorphic with a binary tree. I can prove that every point is reachable by just flood filling the pathways (left image). If all pathways get color, then all points are reachable. Then I can prove that there is exactly one path between any two points by flood filling the background (right image). If there were more than one path between two points, then a background flood fill would leave an island without color.
This process has become second nature to me. The only intellectually difficult part is the planning and planting the deceptions. The rest just flows out of me with very little effort.
To come: How do I make the actual images that serve as subject matter to my mazes?