# Thrice Dice Twice

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

But also six 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