Knutson can still do seven balls by himself, but he is semiretired from juggling now. He keeps 20 or so glittery balls in his desk drawer, partly to show students how to digest complex sets of numbers. His approach is always low tech. While many jugglers rely on computer programs that generate possible throw sequences, such programs tend to propose a slew of aesthetically dull sequences. Knutson prefers to start with nothing but hunches, scratching out little rows of numbers with arrows that describe the balls’ orbits. When pondering the five-ball 3-4-5-6-7 pattern, for instance, he can determine that the first throw, a 3, will land three beats later, as the 6 is being thrown:

3-4-5-6-7

The 4 will stay aloft for four beats, landing as the second 3 is being thrown:

3-4-5-6-7-3

The 5 will float for five beats and land as the next 5 goes airborne:

3-4-5-6-7-3-4-5

In about five seconds, Knutson can see that the 3, 6, 7, and 4 spin in a single orbit, while the 5—always landing back on itself—does its own lonesome, solipsistic dance amid the fray.

In swapping sites, jugglers mutate orbits and follow a simple rule: You can increase the height of one throw in a sequence so long as you equally decrease the height of another throw that lands later—and also pay heed to how much later it occurs. If it’s one throw away, you can add one to the height. In other words, the 4-4-1 mentioned above could be further mutated into a 5-3-1, wherein one ball does an incessant 3 while the other two fly through a 5-1 orbit.

Sometimes, for kicks, siteswappers devise what Knutson calls jugglers-only tricks, improbable sequences whose difficulty is obvious only to the cognoscenti. Knutson’s proudest invention is a five-ball trick, 8-5-7-4-1, that forces the juggler to throw lofty 8s from each hand while continually changing gears to make a wide variety of lesser throws in rapid succession. “I came up with that one to annoy Bruce Tiemann,” Knutson gloats, alluding to a Caltech grad whom many consider the world’s top numbers juggler. “He said, ‘Oh, I can do that,’ and then, after a couple of hours, he gave up in frustration.”

By now, Knutson has exhausted most of the possible juggling sequences. “Either I have them down, or they’re simply beyond me,” he says. “They’re eight- or nine-ball patterns—things I can’t do.” So last fall, along with a Berkeley undergrad named Peter Dolan, he began to consider randomness in juggling.

Pure randomness occurs when a hypothetical juggler is equally likely to make a throw that stays aloft for one, two, three, four, or five beats. Knutson and Dolan want to know what happens when a random juggling pattern becomes increasingly predictable. Is there a mathematical “freezing point” between randomness and predictability, like the phase transition that separates a liquid from a solid?

As they work, Knutson and Dolan are evolving a parameter, q, that denotes where a juggling sequence lies on the continuum between randomness and predictability. If they find, for instance, that juggling routines freeze when q is 32, then computerized juggling programs should be able to use the parameter to weed out what Knutson calls “silly and boring” sequences and focus only on the most delightful. Knutson’s freezing point has a magical aura for siteswappers—like musicians, they yearn for a happy medium between rote predictability and random arm stabbing—and he and Dolan believe they’ve almost found it. “The point we’re

The world record for passing juggling balls was set by Americans Joey Cousin and Bruce Sarafian, who kept 13 balls aloft for 54 catches in 1995 and for 176 catches in 1997. Sarafian also holds the record for the most balls juggled by one person: 10. He once had a car with a Florida license plate that read “11-BALLS.”

seeking does exist,” Knutson says. Mathematicians just need to look a little further. A good theorem, he adds, is like a good juggling routine: “It holds together. It makes sense, and it also delivers pleasant surprises.”