# crazy dice

It is a standard exercise in elementary combinatorics^{} to the number of ways of rolling any given value with 2 fair 6-sided dice (by taking the of the two rolls). The below table the number of such ways of rolling a given value $n$:

$n$ | # of ways |
---|---|

2 | 1 |

3 | 2 |

4 | 3 |

5 | 4 |

6 | 5 |

7 | 6 |

8 | 5 |

9 | 4 |

10 | 3 |

11 | 2 |

12 | 1 |

A somewhat (un?)natural question is to ask whether or not there are any other ways of re-labeling the faces of the dice with positive integers that these sums with the same frequencies. The surprising answer to this question is that there does indeed exist such a re-labeling, via the labeling

$\text{Die}1$ | $=\{1,2,2,3,3,4\}$ | ||

$\text{Die}2$ | $=\{1,3,4,5,6,8\}$ |

and a pair of dice with this labeling are called a set of *crazy dice*. It is straight-forward to verify that the various possible occur with the same frequencies as given by the above table.

Title | crazy dice |
---|---|

Canonical name | CrazyDice |

Date of creation | 2013-03-22 15:01:43 |

Last modified on | 2013-03-22 15:01:43 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 7 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 05A15 |