A colleague Bill Wardlaw (March 3, 1936-January 2, 2013) used to create a “Problem of the Week” for his students, giving a prize of a cookie if they could solve it. Here is one of them.
What is the largest number of regions that a plane is divided into by straight lines in the plane?
Give as a function of and explain why your answer is correct.
What is the largest number of regions that -dimensional Euclidean space is divided into by hyperplanes?
Give as a function of and , and find formulas for , the number of regions that are bounded, and for , the number of regions that are unbounded.
Of course, Explain why these numbers are correct.