Monthly Archives: January 2013

Problem of the Week, #117

Posted on by

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.

Problem of the Week, #116

Posted on by

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 116

Bob invites Joe to see his new program, which prints 2 \times 2 matrices with random integer entries. (These entries are as likely to be even as to be odd.) Joe, being a bit of a gambler, wants to bet Bob a dollar that the next matrix will have an even determinant. Should Bob take the bet? What is the probability that the next matrix will have an even determinant?

Note:
The determinant of the 2 x 2 matrix

A = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)

is given by det(A) = ad - bc.

ADVANCED PROBLEM 116A

What is the probability that the determinant of an n \times n matrix with randomly chosen integer entries is divisible by the prime number p?
(Assume that the residues 0, 1, 2, ... , p-1 are equally likely for each integer entry.)