09/11/2022

In Folha, Viana comments on mathematical intrigues in Las Vegas.

Cassino em Las Vegas | Crédito: Pixabay

Reproduction of Marcelo Viana's column in Folha de S. Paulo.

Standard playing cards contain 52 cards, 13 of each suit. The total number of possible card arrangements is 52 x 51 x 50 x … x 3 x 2 x 1, which we represent as 52! and call 52-factorial. It's a colossal number, with 68 digits. To give you an idea, that's also the estimated number of atoms in the entire Milky Way! That's why guessing cards from a well-shuffled deck is essentially impossible. Unless we know something about their arrangement, of course…

At the beginning of this century, the Las Vegas gambling industry was very worried. A gang had used hidden cameras to film the shuffling machines in the casinos. Played in slow motion, the films revealed a great deal about the order of the cards. The gang used this information to cheat the house… and win. Before they were caught, they had already made millions of dollars at the expense of the casinos. The machines needed to be replaced!

The most common way to shuffle cards is by "cutting" the deck in two (because we humans have two hands…), and mixing those parts more or less randomly. It's not a very good method, because the relative order of the cards in each part remains the same. That's why we need to shuffle several times if we want to really mix them.

The new machine cut the deck of cards into twenty pieces instead of two, reversing the order of the cards in ten of them. Then, the pieces were stacked randomly on top of each other. All of this happened inside the machine, so it couldn't be filmed.

The engineers were very confident that the new machine would work well, but the bosses, burned by past losses, wanted to be sure: would one round of this process be enough to properly shuffle the cards? Knowing that only mathematicians could answer that, they hired Professor Persi Diaconis from Stanford University.

Diaconis' interest was sparked when, at age 13, he met the Scottish magician and computer scientist Alex Elmsley. A year later, he ran away from home to become a professional magician. When he returned to school ten years later, it was to learn the mathematics behind his tricks. Today he is the world's leading authority on the subject.

To read the full text, visit the newspaper's website.