## Contact

### Moscow

College of Science

Mines 321

875 Perimeter Drive MS 3025
Moscow, ID 83844-3025

Phone: (208) 885-6195

Fax: (208) 885-6904

Email: science@uidaho.edu

# Vandal Science News-November 2015

A Newsletter for Alumni and Friends November 2015

## Dean's Message

This autumn has brought many reasons to be proud of our faculty and students at University of Idaho College of Science. From national recognition earned by our faculty members to the achievements of our excellent students, there's much great news to share in this issue. And of course, we'll include a new Vandal Science News Puzzler for your amusement as well.

We invite you to come by the college's 11th annual Student Research Expo on Friday, Nov. 13. Over 40 undergraduate and graduate students will be exhibiting posters of their research projects. And don't forget to drop by the "Idaho Fan Zone" pregame celebration before the football game with Appalachian State on Saturday, Nov. 14. The College of Science will be one of the hosts for this event.

– Dean Paul Joyce

## Vandal Science News Puzzler

We’ll play a simple game for this Puzzler. Suppose we start with a pile of 13 pebbles. Two players take turns making moves, where each move consists of removing 1, 3, or 4 pebbles from the pile. (You must take away at least one pebble in your turn, and you cannot take away any number other than 1, 3, or 4.) The player to remove the last pebble wins. There is an easy-to-describe winning strategy for this game that can be discovered by some analysis. Figure out the strategy and then answer this question: if you are the first player, what should be your initial move?

## Solution

Your move should be to remove four pebbles, leaving 9 – here's why.

The status of the game at any time is indicated by the number of pebbles remaining, which can be anything from 0 to 13. Each of our moves will land on one of those numbers, and we'll keep track of the ones we want to land on with a pink circle.

• Now clearly we want to land on zero – that's how we win. We can do that if our opponent lands on either 1, 3, or 4 – so we'll put green squares around those – that's the first row in the diagram at right.
• Since position 2 isn't spoken for yet (that is, it isn't one we want to make our opponent land on), it must be one that we want to land on. (And notice: if we land on 2, then our opponent's only choice for his move is to remove one pebble, which means we would win in our next move!) But we can land on 2 for our move if we force our opponent to land on either 3, 5, or 6. Now 3 already has a green square, but we'll add squares to 5 and 6, which gives us the 2nd row.
• This means we want position 7 (the lowest available “unclaimed” spot) as one of our pink circles. This means we put green squares on 8, 10, and 11 as positions we want to force for our opponent. This gives the 3rd row.
• Position 9 is then left as one of our desired landing spots, and this then adds green squares to 12 and 13. But more importantly, since we know that landing on 9 is a winning move, we can see that our first move should be to go to that position by removing four pebbles!

So the game strategy becomes this: we always move to land on one of the pink circle positions. Our opponent will then have no choice but to land on a green square position, which in turn lets us get back to a pink circle. Since we're the only player that can get to a pink circle position, we're the only one who has a chance to win!

Pebble Puzzler Solution

### Correct Solvers

• Fred Burton (Mathematics, 1968)
• Tim Householder (Mathematics, 2002)
• Chris Marx
• Diane McGarry
• Leland Ogren (Chemistry, 1974)
• Greg Stenback (Geological Engineering, 1985; MS Statistics, 1987)

## Contact

### Moscow

College of Science