Five Clean Throws and a Sticky Note That Says 423

Three beanbags arrived by courier this morning, and I have already dropped them a conservative four hundred times. My left shoulder aches. There are two bags under the couch that I cannot reach without moving the couch. And I have abandoned lunch to write this down, because somewhere between drop two hundred and drop three hundred I discovered that juggling has a notation, and I cannot keep it to myself for another hour.

Back up. Saturday, chess tournament, that dead half-hour between rounds when everyone drifts off to lick their wounds. A fellow in the corner had just lost a winning endgame — hung a rook, the way you do when you’ve been staring at sixty-four squares for three hours — and instead of stewing he pulled three beanbags out of his jacket and started throwing them in a lazy figure-eight. Said it cleared the board out of his head faster than analysis ever did. He tossed them to me. I produced zero clean throws and spent an undignified minute retrieving bags from under a neighbouring table. Ordered my own set before I’d even left the parking lot.

The first thing everyone gets wrong

You throw forward. Every beginner does. The bags drift away from you, you lurch after them, and within four throws you’ve migrated across the room like a man chasing his own hat down a windy street. The fix is almost insultingly simple: throw straight up, in a flat vertical plane parallel to your chest, as if you’re working inside a pane of glass. Each bag peaks at about eye level and drops into the opposite hand. Once I imagined the glass, my catch rate roughly tripled.

The other instinct to fight is the urge to make a circle — one hand always scooping, always throwing high, bags going round like a waterwheel. It looks like “real” juggling. It’s called the shower, and it is genuinely harder than the pattern beginners should actually learn, the cascade, where the hands alternate and each throw arcs under the incoming one. The cascade’s asymmetry is the whole point. Even numbers of objects fountain in two separate columns; odd numbers cross. Three crosses. Three is where you start.

Then I found out it’s math

Here is what sent me to write this mid-drop. There is a notation for juggling patterns called siteswap, invented independently by three different groups of people around 1985 — one at UC Santa Cruz, one pair of Americans, and Colin Wright’s crowd in Cambridge — who all arrived at the same idea within a couple of years, apparently without knowing about each other.

It works like this. Each throw is a number: how many beats until that same ball is thrown again. A plain three-ball cascade is just 3. Every throw is a 3. The whole pattern collapses to a single digit. The average of the digits always equals the number of balls you’re holding — so 51 is a three-ball pattern too (a 5 thrown high while a 1 zips straight across), and so is 423, which hides a 2: a beat where one hand simply holds a ball and does nothing while the other hand works. Odd numbers cross to the far hand, even numbers come home to the same one. You can check whether a sequence is even possible with modular arithmetic — add each throw to its launch time, take it mod the period, and if any two collide, two balls try to land in the same hand at the same instant and the pattern is a lie.

People have discovered tricks on paper this way — 645, 552 — patterns nobody had thrown, found by writing valid numbers and then working out what they’d look like in the air. A skill with a debugger. I did not expect that.

The connection I keep circling: Claude Shannon, who wrote the 1950 paper that taught computers to play chess, also built juggling machines and proved a theorem relating flight time, dwell time, and the number of balls. The same man, both ends of my Saturday.

I still can’t cascade. Five clean throws is my record, and it happens by accident, never on demand. Learning the padlock taught me that the hands calibrate on their own schedule, without consulting me — pin three binds first, and one day your fingers simply know it. My hands haven’t caught up to the notation yet. But 423 is written on a sticky note above my desk, and I understand exactly what it wants me to do. That gap — between reading the pattern and throwing it — is the entire game right now, and I am delighted to be losing it.