### Game #1

Two players take turns breaking a chacolate bar (rectangle-shaped consist of squares). The last to break a piece wins the game.

Design the strategy.

#### Solution

Each time the bar is broken, **total number of pieces increase by 1**. Suppose there’re even number of squares, 1st player wins regardless of breaking strategy. And vice versa.

### Problem #2

75 teams took part in a competition where teams met 1-on-1. Each time the defeated team drops out.

How many meets are needed to before one team is declared a winner?

#### Solution

Each game will eliminate 1 game, so it needs 74 games.

### Splitting Piles

Given a random number of items in a pile. Ask an audience to split a pile into two piles, multiply the numbers of items in the two new piles and keep adding the results. The process stops when there is no pile with more than 1 chip.

For example, let start with 9 chips:

Piles | Which is broken | What's added | Total |

9 | 9 | 3*6 | 18 |

3,6 | 3 | 1*2 | 20 |

1,2,6 | 6 | 3*3 | 29 |

1,2,3,3 | 3 | 1*2 | 31 |

1,2,1,2,3 | 2 | 1*1 | 32 |

1,1,1,1,2,3 | 2 | 1*1 | 33 |

1,1,1,1,1,1,3 | 3 | 1*2 | 35 |

1,1,1,1,1,1,1,2 | 2 | 1*1 | 36 |

1,1,1,1,1,1,1,1,1 | - | - | - |

Before the audience told the final number, you immediately guess it’s 36. How did you do it?

#### Solution

The result does not depend on how the piles are split; but only on the initial size of the very first pile. **Answer is always N(N - 1)/2**.

This can be proved by mathematical induction.