DAILY COUGAR PUZZLE

 

DEPARTMENT OF MATHEMATICS

 

 

The puzzle appears in the University of Houston Daily Cougar for two days, twice a month, during September, October and November during the fall semester and February, March and April during the spring semester.

 

RULES FOR THE PUZZLE CONTEST

 

( 1.)    Anyone may submit a solution to a puzzle.

 

( 2.)    A solution may only be submitted either by

(a.)     email to puzzle@math.uh.edu

           or       

                  (b.)    in person, at the reception desk of the Department of Mathematics in a properly marked                                  envelope.

 

( 3.)    In order to be eligible for the contest, a submission must be received no later than five (5) days after the first appearance of the puzzle in the Daily Cougar.

 

( 4.)    All solutions submitted must have the full name of the submitter and an email address to contact the            submitter.

 

( 5.)    All timely submitted solutions will be graded by the Mathematics Department.

 

( 6.)    The University of Houston undergraduate student submitting the best correct solution will be awarded a $50 gift certificate by the Mathematics Department.

 

( 7.)    The winner of the gift certificate will posted in the Daily Cougar and will also be notified by email.

 

( 8.)    All puzzle decisions as to best and correctness are made by the Mathematics Department and the decisions are final.

 

( 9.)    A winner, if any, and a solution for each of the puzzles will be posted at this Mathematics Department Cougar Puzzle web site.

 

(10.)   A record is kept of the solvers for each month and the top undergraduate problem solver for the entire academic year will receive a $250 gift certificate from the Mathematics Department and is qualified to attend the US National Collegiate Mathematics Championship. The 2007 USNCMC was held in San Jose, California this past August.

 

           Some of the puzzles are supplied by: ÒThe Problem Solving CompetitionÓ.

 

 

                                  DAILY COUGAR SOLUTIONS

 

DC PUZZLE NUMBER 1

 

Winner: Ms. Melody Lam

 

                                  PRIME NUMBERS

 

Determine if there is a positive integer n such that both 2n + 1 and 2n – 1 are prime numbers. If there is such a number, then determine all such. Carefully justify your answer.

 

 

SOLUTION

 

Note that 2n is not divisible by 3 and if x, y and z are three consecutive positive integers, one of x, y or z is divisible by 3. Thus either 2n + 1 or 2n – 1 must be divisible by 3. Further, if both of these numbers are to be a prime number, then one of them must be 3.

 

If  2n – 1 is 3, then 2n + 1 is 5 (n = 2).

 

If  2n + 1 is 3, then 2n – 1 is 1 (n = 1).

 

But the number 1 is not a prime number.

Therefore 2 is the only possible choice for the integer n.