Opinion – Marcelo Viana: Mathematical intrigues in Las Vegas

Regular decks contain 52 cards, 13 of each suit. The total number of possible orderings of the cards is 52x51x50x…x3x2x1, which we represent by 52! and we call it 52-factorial. It is a colossal number, with 68 digits. To give you an idea, this is also the estimated number of atoms in the entire Milky Way! This is why guessing cards from a well-mixed deck is essentially impossible. Unless we know something about their ordering, of course…

At the beginning of this century, the Las Vegas gaming industry was very concerned. A gang had used hidden cameras to film the shuffling machines in the casinos. Played in slow motion, the films made it possible to discover a lot about the order of the cards. The information was used by the gang to play against the bank… and win. Before they were caught, they had already made millions of dollars at the expense of the casinos. The machines needed to be changed!

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

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

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

Diaconis’ interest was piqued when, at the age of 13, he met 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, he went to learn the math behind his tricks. Today he is the world’s leading authority on the subject.

Diaconis’s conclusion about the new scrambling machine was not what the Las Vegas businessmen and their engineers had expected. That’s a topic for next week.