Shannon’s Juggling Theorem

Claude E. Shannon is best known for his 1948 paper “A Mathematical Theory of Communication” in which he created the field of Information Theory, but he had many other important contributions in diverse areas.

What is lesser known is that he had a keen interest for gadgets and devices of all sorts and he invented some quite humorous ones, like “The Ultimate Machine” (also known as “The Most Useless Machine Ever”):

Shannon was also an accomplished juggler. He came up with the following elegant theorem, known as

Shannon’s Juggling Theorem


F is the time a ball spends in the air (Flight)

D is the time a ball spends in a hand (Dwell), or equivalently, the time a hand spends with a ball in it

V is the time a hand spends empty (Vacant)

N is the number of balls

H is the number of hands

The theorem can be derived by looking at a complete juggling cycle first from the perspective of the ball, then from the perspective of the hand, then equating the two times. This is an application of one of the most useful general tricks in combinatorics: double counting. You count/measure something in two different ways (in this case the juggling time), and use the fact that the two results have to be equal.

We can read out from the theorem some obvious facts, such that if you throw the balls higher (increase F) then V will also increase (your hands will be empty for longer). If you increase D at the expense of F and V, until they become zero (you keep holding the balls in your hands), N and H have to be equal (one ball in each hand). No surprises here, except to note that the theorem assumes that there is at most one ball in one hand at a time, so it does not apply to multiplex patterns in which several balls are simultaneously held in the same hand (we would need separate Ds for hand and ball to fix this, but the simplicity of the theorem would be lost).

What if you want to juggle more balls (increase N) but you cannot change F, V or D (you cannot juggle any faster or throw the balls any higher)? No problem, just increase the number of hands (H). One way to achieve that is by becoming more social.

Further links:

The Science of Juggling

A personal tribute to Claude Shannon


#1 Cathy on 12.22.10 at 11:04 pm

Hi, thanks for the video link, its hilarious. I stumbled on this page as I am studying the mathematics of juggling at university. If you or anyone could lead me to an understandable full proof of Shannon’s juggling theorem that would be really useful.

#2 lkozma on 12.23.10 at 7:40 am

Hi Cathy,
there’s not much more to the theorem than what I wrote in the 4th paragraph. There’s a diagram at the bottom of the “personal tribute” link which could make things clearer.

Maybe it’s easier to see if you write it in the form: (f+d)/n = (v+d)/h

The left part is the total time from the perspective of the balls: a ball is either in the air, either in a hand, and we have to divide by n, since we counted each ball.

The right part is the exact same total time, from the viewpoint of the hands: a hand is either empty, or has a ball in it, and we added this up for all hands, thus we have to divide by h.

#3 Greg Wetzel on 11.10.11 at 10:40 pm

I like juggling and applying math with juggling is great stuff as what Shannon has done with his theorem. Do you find the equation helps a juggler understand juggling better? I could see how this theorem may help with team juggling. With a juggler juggling on his or her own the theorem does not seem to be of much use. I have not seen many jugglers use this equation. Nevertheless this equation is still very neat. Siteswap is used much more with jugglers.

#4 lkozma on 11.11.11 at 7:43 am

Greg: I agree completely, this is more like a fun fact, an equation which is surprisingly neat, but not something to make serious use of.