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.*

Wed Jan 6 11:37:02 EST 1999