# Problem of the Week: Futility Rocks

Editor’s Note: The Department of Mathematics at Gettysburg College hosts a problem of the week challenge to determine each semester’s Paul Mugabi problem-solving award recipient(s).  Each week’s entries are scored by a faculty judge, and winner(s) from each week will receive a Problem Of the Week (P.O.W.) button.  The Gettysburgian is not involved in or responsible for accepting or evaluating students’ submissions to this contest.

THE RULES:

The contest is open to all Gettysburg College students. Up to three people may work together on a submission. Make sure your name is on your submission and that any sources are properly cited. Send solutions to bkennedy@gettysburg.edu. This problem was posted on Friday, April 15 and solutions are due on Friday, April 22 by 5:00 p.m.

This is the last Problem of the Week of the semester.

THE PROBLEM:

I have just been hired to work in Earl Eccentric’s rock garden, where there is one pile containing A rocks and another pile containing B rocks.

Here is my job. At the beginning of every hour, I will calculate the total number T of rocks in the two piles together, and

• if T is even, I will move T/2 rocks from the larger pile to the smaller pile;

• if T is odd, I will choose one rock (from either pile) to throw away into the bushes.

My task will be complete when the two piles have the same number of rocks; after that, I do not have to move any more rocks.

QUESTION:

For which values of A and B is it possible for me to complete my task? Answer, and explain.

Author: Gettysburgian Staff