I found the solution to that first maze but didn't find the rules. Most of my guesses were wrong. From the solution I gather you don't have to travel every path, and you can cover the same path more than once. What I don't understand is why they went the way they did when you could enter on the right, drop down, over two, up three, hang a left and on till morning... I mean out.
I just noticed they did pass by every square on at least one side.