Four, four, one is three balls
Jugglers write a pattern down as a list of numbers. A throw is the number of beats the ball stays up before it is thrown again, and the hands take turns, one beat each. Three balls in the ordinary cascade are thrown 3, 3, 3, … for ever, which is written 3. That is the whole notation, and two facts fall out of it. The average of the list is always the number of balls. And a list can be juggled exactly when no two throws come down on the same beat — which the average cannot tell you on its own.
Write a pattern
Type a list of throws, or pick one. A letter is a throw of ten or more: a is ten, f is fifteen. The page opens on 441 because the title does; it is not special, and every pattern below works the same way.
Follow one colour and you follow one ball. Every beat with a ball on it throws it again, so each colour runs from arc to arc without a gap and never ends — and the dashed upright, wherever you put it, cuts exactly one arc of each colour.
The same pattern, thrown
Odd throws cross to the other hand and even ones come back to the hand that threw them. A ball is held for half a beat between landing and being thrown again; the heights go as the square of the time in the air, the way a thrown thing does. A 2 is usually just held rather than thrown, and a 0 is a hand with nothing in it.
Why the average is the number of balls
Stop the clock between two beats. Every ball is somewhere on its way from one throw to the next — in the air, or in a hand about to throw it — so it is on exactly one arc. The number of balls is the number of arcs crossing that moment.
Now count ball-beats over one pass through the pattern, n beats long. With b balls, each of the n moments has b arcs across it: b × n. From the other side, a throw of t is across t of those moments; and because the pattern repeats, the moments a throw from this pass spends in the next are paid back exactly by the last pass's throws spending theirs in this one. So the throws account for t₁ + t₂ + … + tₙ. The two counts are the same count: t₁ + … + tₙ = b × n, and the average is b.
The page does not take that on trust. It counts the balls both ways for every pattern it builds — by the average, and by standing at a moment and counting the throws that cross it — and the check at the foot says whether they ever disagree.
The average is not enough
432 averages three. It cannot be juggled: the 4 thrown on beat 1, the 3 on beat 2 and the 2 on beat 3 all come down on beat 5, and beats 6 and 7 get nothing. 423 is the same three throws in another order and it juggles. A whole-number average is necessary — a list whose average is not whole cannot be juggled at all — and not sufficient.
What is surprising is how little more you need. Any list of throws whose average is a whole number can be put in some order that juggles. That follows from a theorem about adding up on a clock face, which Marshall Hall proved in 1952 in a paper that has nothing to do with juggling, and this page does not prove it. It checks it: every list of up to six throws, none higher than nine, whose average is whole.
Every trick for a few balls
Each of these repeats within three throws, and each is written the way jugglers write it, starting from its highest throw — 441, never 414 or 144, which are the same trick joined at a different place. Pick one to load it above.
How many patterns there are
Count every list of n throws that juggles b balls. Every list, not every trick: 441, 414 and 144 are three lists here, and 33 is a list of two. The page counts them by building them. A list that juggles is two choices — which beat each throw comes down on, which must be a shuffle of the beats, and how many extra passes each throw spends in the air — and the average fixes how many extra passes there are to share out. So the table is every shuffle of up to six beats, each sharing out what is left, for up to five balls.
That is a theorem too — Joe Buhler, David Eisenbud, Ron Graham and Colin Wright proved it in 1994 — and the table is the page arriving at it rather than quoting it. Read the formula the other way round and it says something odd: there are exactly as many lists of n throws juggling at most b balls as there are strings of n digits running from 0 to b.