Passe-dix
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Do you know the game of passe-dix? Three dice are thrown. If the sum shown on the upper faces of the three dice exceeds ten, the thrower wins; otherwise he loses. The game is ancient and has given rise to many problems. The 19th-century French writer Villiers de L'Isle-Adam regretted that "the passe-dix dice-cups rolled dice on stakes capable of starving provinces."
You could carry out long combinatorial calculations to work out the probability of each roll, but it is cleverer to note that the sum of the digits on two opposite faces of a die equals 7. So across the three dice, the total of the digits on opposite faces is 21.
To every roll of three dice whose sum is less than 10 there corresponds, on the symmetric faces (the bases on which the dice rest), a sum greater than 10… and conversely. So the probability of winning is 1/2. Quite simply!
The Grand Duke of Tuscany was puzzled that the sum 10 came up more often than the sum 9, even though each has six ways of occurring (10 = 6 + 3 + 1 = 6 + 2 + 2 = 5 + 4 + 1 = 5 + 3 + 2 = 4 + 4 + 2 = 4 + 3 + 3, and 9 = 6 + 2 + 1 = 5 + 3 + 1 = 5 + 2 + 2 = 4 + 4 + 1 = 4 + 3 + 2 = 3 + 3 + 3). Galileo solved the problem by calculating that 9 can occur in twenty-five different ways, whereas 10 occurs in twenty-seven different ways, out of the two hundred sixteen possible combinations. It is therefore easier to get 10 than 9.
