# Winning Strategy

## Objective

Analyze a game and develop a winning strategy.

## Difficulty

Procedure: Easy

Concept: Hard

## Concept

In this particular game, there are two players who take turns counting, starting at 1. A player may only count in increments of 1 or 2. The player who says 21 loses. Here are two possible outcomes demonstrating how the game is played.

Outcome 1 | Outcome 2 | |

Player 1 says | 1,2 | 1 |

Player 2 says | 3 | 2 |

Player 1 says | 4,5 | 3,4 |

Player 2 says | 6,7 | 5,6 |

Player 1 says | 8 | 7 |

Player 2 says | 9 | 8 |

Player 1 says | 10,11 | 9,10 |

Player 2 says | 12 | 11,12 |

Player 1 says | 13,14 | 13 |

Player 2 says | 15,16 | 14,15 |

Player 1 says | 17 | 16 |

Player 2 says | 18 | 17 |

Player 1 says | 19,20 | 18 |

Player 2 says | 21 | 19,20 |

Player 1 says | 21 | |

Result | Player 1 Wins | Player 2 Wins |

This is a finite-length two-player game with two possible outcomes: either player 1 or player 2 wins.

## Hypothesis:

What patterns do you notice that force your opponent to lose regardless of his or her choice of counting?

## Materials:

- Paper and pencil
- Computer (if you're up to programming a winning algorithm!)

## Procedure:

- Find a friend and play the game.
- Record what happens at each turn.
- Based on the outcomes, develop a strategy.
- Play the game again to test your strategy.
- Repeat steps 1-4 if necessary

Remember that you are aiming to develop a *foolproof* strategy. This means that you will win no matter what your opponent does if your strategy is followed correctly.

## Analysis

Once you construct a step-by-step winning strategy, try to generalize the strategy. What happens if the game ends at a different number (22 instead of 21)? What if the players were allowed to count in increments of 1,2, or 3? Try to mathematically relate the patterns you noticed to these different parameters.

HINT: Look into the modulo operator (%). This gives the remainder of two numbers (i.e., 9 % 4 = 1).

## Conclusion

Did you hypothesize the winning strategy? If not, what are some properties of the game that you can define? In other words, what did you notice that gives you a losing strategy?

## Extensions

Once you've become a master of the game, try developing a computer program where the user can define his or her own parameters and play the game. What happens if there are more than two players?