# The Coin Game Caper

## Objective

To examine a simple game and test it for fairness.

Procedure: Easy

Concept: Hard

## Concept

How do you define fairness? In the context of a game, does that definition change? In this experiment, we'll be analyzing a very simple coin game to see whether or not it is "fair." In this particular game, there are 2 players that take turns removing coins from the ends of a line of coins arranged randomly. The player with the most money when all the coins have been taken wins. The picture below is an example of how a game would begin:

Player 1 would begin by removing a coin from either end of the line. In this case Player 1 chose to remove a coin from the left:

Player 2 would now choose to remove a coin from either end of the line. If Player 2 chose to remove the quarter from the right, the game would look like this:

The game continues in this fashion until all coins have been selected. Both players add the values of their coins and whoever has more money at the end wins.

• Coins

## Hypothesis

Is there a strategy that exists that will guarantee that the first player will at least draw with the second player?

## Procedure

1. Take coins of various denominations and arrange them randomly in a row.
2. Make a note of the order of the coins so that the game can be played multiple times.
3. Play the game a couple of times.
4. Based on the outcomes, attempt to come up with a strategy for the game that guarantees a win.
5. Play again with the same arrangement of coins but this time using the strategy.
6. Repeat Steps 1 through 5 with different arrangements of coins.

## Analysis

Are your pattern recognition skills up to snuff? Start by choosing two or three very simple strategies. If those don't work, increase the complexity. Be sure to run each strategy through the procedure above a sufficient number of times, and remember it's easier to prove a losing strategy than a winning one.

## Extensions

Try a different game! Either use one that you've seen before or just make one up. The simpler the rules, the easier it should be to find a winning strategy (if one exists). If possible, a computer simulation might help to test a strategy for every possible arrangement of coins.