What is the maximum number of regions you can divide space up into using 7 planes? The investigation of this type of problem, especially highlighting techniques of problem solving, is the subject of an old film by George Polya entitled ``Let us Teach Guessing,'' available from the Mathematical Association of America. I saw this film with my class in high school, and it made a profound impression on me.
When thinking about this problem, consider simpler problems (use fewer planes) and analogous problems (dividing a plane up by lines or dividing a line up by points). Try to relate the results.
In my opinion, no teacher should be without Polya's book How to Solve It. Another good book of his is Mathematical Discovery. Two more advanced books on problem solving by Polya are Mathematics and Plausible Reasoning, Volume I: Induction and Analogy in Mathematics and Mathematics and Plausible Reasoning, Volume II: Patterns of Plausible Inference.