Consider the diagram:
The goal is to draw three paths that each must start at the boundary of the image satisfying these conditions:
No path crosses itself.
Each path crosses the other two paths exactly once.
The seven pieces (faces) must have equal areas.
Here is an example that satisfies the first two conditions but not the last just to give an idea of exactly what the question is asking:
I have been told there are two solutions, but I have not yet found them. Of course, if there is a solution, then you could stumble upon it just by guessing. However, that takes a long time so what I'm really trying to determine is this - is there a systematic procedure that can be used to determine a "correct" path?
Here are some things that I've determined that could help:
a) Each "face" of the graph has five squares.
b) In order to make each path cross the other two exactly once, there must be one face in the center, and the other six faces must all be touching the outside boundary.
c) Using fewer "bends" or "turns" for each path is ideal because the more times you bend, the less places that are available for one path to cross another one without overlapping it.
d) There is a finite number of pentominoes that can be used - moreover, there is a finite number of pentominoes that can be used for the center "face".
e) If the method to figure this out is to draw one path at a time, the first path much section the diagram into 15 squares on one side of the path and 20 on the other side. Furthermore, the next path drawn must cut the 15 squares (from the first path) into 5 squares and 10 squares, and it must cut the 20 squares (from the first path) into 10 squares and 10 squares - this ensures that if the first path were removed, the second path will have divided the diagram into 15 squares and 20 squares.
I appreciate any suggestions - again, I'm not really looking for what the solution is, but rather how the solution can be found.