The Grand Duke's Dice Dilemma
The Grand Duke of Tuscany noticed something odd at the gaming tables: rolling three dice and adding them up, a total of 10 turns up more often than a total of 9 — even though there are exactly six different combinations of three numbers that sum to 9, and exactly six that sum to 10. He asked Galileo to explain the discrepancy. Where does the 'six equals six' reasoning go wrong?
Reveal the answer
Not every combination is equally likely, because the three dice are distinguishable: {1,2,6} can land in 6 different orders, but {3,3,3} can only land one way. Counting every ordered outcome among the 216 possible rolls, 9 is made 25 ways and 10 is made 27 ways — so 10 truly is more common. Galileo's short 1620 paper on this is one of the earliest rigorous applications of counting principles to probability, decades before Pascal and Fermat.