Department of Mathematics Gettysburg College.

## The Math Problem of the Week Contest

*Spring 2022, Number 11*

### 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.

Weekly winners will receive a POW victory button, and the best-performing students of the semester will receive the Paul Mugabi problem-solving award.

**Problem posted**: Friday, April 15

**Solutions Due**: Friday, April 22 5:00 p.m.

**Send submissions to:** bkennedy@gettysburg.edu.

**This is the last P.O.W. of the semester.**

#### Futility Rocks

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.