In this video James Grime examines the “challenging” math problem given in the movie Good Will Hunting and points out that it is not actually all that challenging. Unfortunately he is pressed by the person interviewing him to give all of the solutions to the dots and lines problem given.
This problem could easily be extended to be more open-ended simply by leaving the number of dots open. Are there any patterns when you generate diagrams with 2 dots, 3 dots, 4 dots, 5 dots, and so on? What kinds of diagrams are essentially the same (homeomorphic)? What kinds of diagrams cannot be made more simple without changing the character of the diagram (irreducible)?
Math Thinking 
Reblogged this on The Musical Mathematician and commented:
So, one of the things I’m doing with my class right now is a monthly math challenge. This month we are working on the Five Room House Problem. We spent some time working on the problem as a class, and then I posted it in the room, leaving space for students to add ideas. I plan to use some of this information in the development of their Math Journal Notebooks. I think that this problem will be next months.