# King of chance won't leave it to one shuffle Fill's best deals follow full mix-up

So you shuffle the cards three, four, maybe five times. How much have you mixed them up?

Not enough, says Jim Fill, an associate professor of mathematical sciences at the Johns Hopkins University. You'd better shuffle them seven times. Or better yet, eight, nine, even 10 times.

The good-natured Mr. Fill, 37 and fresh-faced, knows this because he has studied card shuffles for five years.

He gives talks -- colloquiums, they're called -- entitled, "The Mathematics of Card Shuffling and a Self-Organizing List

Scheme."

Don't go to one unless you understand such words as asymptotics, eigenvalues and stochastic characterizations.

At a recent colloquium at the Applied Physics Laboratory in Laurel, the 94 engineers and computer wonks nodded and asked intelligent questions.

Mr. Fill did bring along his deck of cards, though, and several days later in his office at Hopkins he laid them on the table.

"I've always been good at math," he said. "I was good at arithmetic in kindergarten."

He wrote his doctoral thesis at the University of Chicago on the question: How many times do you have to flip a coin before the percentage of heads and tails gets close to half?

You'd think there was a simple answer to that.

"It's fairly technical to say in what sense I gave that answer," Mr. Fill said.

Bet your last poker chip he didn't flip a dime a thousand times to find out. No, he wrote 100 pages on the topic and got published.

Then, while teaching at Stanford University, Mr. Fill heard a fellow professor in the statistics department, Persie Diaconis, speak on the subject: How long do you have to mix things up to get them really mixed up?

That got Mr. Fill, who had played a little poker in his day, interested in card shuffles.

He and Mr. Diaconis eventually developed a theory they called "Strong Stationary Duality."

You'd think there was NOT a simple explanation to that.

"The theory gave us, well," Mr. Fill said, trying to figure out the simplest way of saying this. "There's really no absolute answer to any of this, because it's a matter of how close is close."

Exactly. But the important thing here is that the two brilliant probabilists/mathematicians illustrated their theory with a card shuffle.

This one we can all understand. Take the top card off the deck and stick it at random into the deck. Do this about 300 times and the deck will be reasonably mixed up.

Mr. Fill realized that illustrating a theory this way had one drawback.

"Nobody actually shuffles a deck by taking one card off the top and sticking it in the middle."

Many players in neighborhood poker games use the riffle shuffle. That's the one that requires at least seven shuffles to mix up the cards.

So is Mr. Fill some sort of wizard card player?

"I used to play poker for some pretty serious stakes," he said. "But I haven't played in maybe 10 years.

"I was good at the mathematics of it, but I was not so good at poker faces. When I saw a great hand, I wanted to be happy about it."

So he contents himself with the intellectual challenge of it all, the interplay between theory and application.

Mr. Fill also teaches at the Baltimore campus.

He is currently the enrollment king of his department, instructing about 300 students in two classes, introductory statistics and introductory probability.

"We're not doing card shuffles," he said a little defensively.

But shuffles are, he pointed out, very good illustrations of very serious mathematics.

"That's a good thing about my job," he said.

"It's actually possible, or closer to possible than in most mathematical disciplines, to tell people on the street what it is I do."

The technical name of what he does is: "The Study of Rates of Convergence to Stationarity for Markov Chains."