A once very difficult but now very simple problem in probability from Ian Stewart’s *Do Dice Play God?* (2019):

For three dice [Girolamo] Cardano solved a long-standing conundrum [in the sixteenth century]. Gamblers had long known from experience that when throwing three dice, a total of 10 is more likely than 9. This puzzled them, however, because there are six ways to get a total of 10:

1+4+5; 1+3+6; 2+4+4; 2+2+6; 2+3+5; 3+3+4

Butalsosix ways to get a total of 9:

1+2+6; 1+3+5; 1+4+4; 2+2+5; 2+3+4; 3+3+3

So why does 10 occur more often?

To see the answer, imagine throwing three dice of different colors: red, blue and yellow. How many ways can you get 9 and how many ways can you get 10?

Roll | Total=9 | Dice #1 (Red) | Dice #2 (Blue) | Dice #3 (Yellow) |

01 | 9 = | 1 | 2 | 6 |

02 | 9 = | 1 | 3 | 5 |

03 | 9 = | 1 | 4 | 4 |

04 | 9 = | 1 | 5 | 3 |

05 | 9 = | 1 | 6 | 2 |

06 | 9 = | 2 | 1 | 6 |

07 | 9 = | 2 | 2 | 5 |

08 | 9 = | 2 | 3 | 4 |

09 | 9 = | 2 | 4 | 3 |

10 | 9 = | 2 | 5 | 2 |

11 | 9 = | 2 | 6 | 1 |

12 | 9 = | 3 | 1 | 5 |

13 | 9 = | 3 | 2 | 4 |

14 | 9 = | 3 | 3 | 3 |

15 | 9 = | 3 | 4 | 2 |

16 | 9 = | 3 | 5 | 1 |

17 | 9 = | 4 | 1 | 4 |

18 | 9 = | 4 | 2 | 3 |

19 | 9 = | 4 | 3 | 2 |

20 | 9 = | 4 | 4 | 1 |

21 | 9 = | 5 | 1 | 3 |

22 | 9 = | 5 | 2 | 2 |

23 | 9 = | 5 | 3 | 1 |

24 | 9 = | 6 | 1 | 2 |

25 | 9 = | 6 | 2 | 1 |

Roll | Total=10 | Dice #1 (Red) | Dice #2 (Blue) | Dice #3 (Yellow) |

01 | 10 = | 1 | 3 | 6 |

02 | 10 = | 1 | 4 | 5 |

03 | 10 = | 1 | 5 | 4 |

04 | 10 = | 1 | 6 | 3 |

05 | 10 = | 2 | 2 | 6 |

06 | 10 = | 2 | 3 | 5 |

07 | 10 = | 2 | 4 | 4 |

08 | 10 = | 2 | 5 | 3 |

09 | 10 = | 2 | 6 | 2 |

10 | 10 = | 3 | 1 | 6 |

11 | 10 = | 3 | 2 | 5 |

12 | 10 = | 3 | 3 | 4 |

13 | 10 = | 3 | 4 | 3 |

14 | 10 = | 3 | 5 | 2 |

15 | 10 = | 3 | 6 | 1 |

16 | 10 = | 4 | 1 | 5 |

17 | 10 = | 4 | 2 | 4 |

18 | 10 = | 4 | 3 | 3 |

19 | 10 = | 4 | 4 | 2 |

20 | 10 = | 4 | 5 | 1 |

21 | 10 = | 5 | 1 | 4 |

22 | 10 = | 5 | 2 | 3 |

23 | 10 = | 5 | 3 | 2 |

24 | 10 = | 5 | 4 | 1 |

25 | 10 = | 6 | 1 | 3 |

26 | 10 = | 6 | 2 | 2 |

27 | 10 = | 6 | 3 | 1 |