Problem of the Week, #117

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.

PROBLEM 117

What is the largest number of regions r(n) that a plane is divided into by n straight lines in the plane?
Give r(n) as a function of n and explain why your answer is correct.

PROBLEM 117A

What is the largest number of regions r(n, d) that d-dimensional Euclidean space is divided into by n hyperplanes?
Give r(n, d) as a function of n and d, and find formulas for b(n, d), the number of regions that are bounded, and for u(n, d), the number of regions that are unbounded.
Of course, r(n, d) = b(n, d) + u(n, d). Explain why these numbers are correct.

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