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