The Mathematics of Passing - theJugglingCompany.com

Objects minimum for a 2-person passing pattern

3 objects per person, shared between two jugglers - the equivalent of two independent cascades that exchange objects on a schedule

1995

Prechac notation published

by Martin Probert and Christophe Prechac, extending siteswap to multi-juggler patterns

~30

Named 2-person 6-club patterns

including the 3-count, 4-count, chocolate bar, and their variations - each a distinct cycle in the passing state graph

∞

Valid 3-person passing patterns

the state space expands exponentially with juggler count - most have never been performed

Two-person passing looks like chaos. It is not. The space between two passing jugglers is a directed graph being traced in real time: every arc obeys a specific trajectory, every prop arrives at a specific hand at a specific moment, and the whole structure is determined by an extended form of siteswap.

When juggling is shared between multiple jugglers, the mathematics extends from individual sequences into a higher-dimensional structure - a shared state space where each person’s throws affect what the other person can throw. This is passing siteswap, formalised by Probert and Prechac (1995) and built on the foundational solo theorem of Buhler, Eisenbud, Graham and Wright (1994).

From solo to duet: the shared state

In solo siteswap, the state at any moment is a description of which future beats already have a ball scheduled to land. For a 3-ball juggler, this is typically a 3-bit or 5-bit string depending on the maximum throw value.

When two jugglers share objects, the state must encode the future landing schedule for both jugglers simultaneously. A pass - an object thrown from juggler A to juggler B - changes both A’s state (one object leaves A’s future schedule) and B’s state (one object enters B’s future schedule).

The constraint from solo siteswap extends: every beat must have exactly the right number of objects landing. The modular arithmetic check now applies across the combined schedule of both jugglers.

Prechac notation

Standard siteswap cannot represent passing patterns because it does not encode which juggler catches each throw. Prechac notation (developed by Christophe Prechac and Martin Probert in 1995, formalized by Probert) extends siteswap by adding a letter to mark passes.

In Prechac notation for two jugglers:

Each throw value v represents the number of beats before landing, as in siteswap
A suffix p marks a pass - the object goes to the other juggler
The two jugglers’ sequences are written separated by a pipe: A | B
Both sequences must have the same period (they share the beat grid)

The validity condition extends the siteswap theorem: for each juggler’s sequence independently, and for the combined sequence across both jugglers, the modular distinctness condition must hold.

Example: the 3-count

The most common 2-person club passing pattern. Sequence: 3 3 4p | 3 3 4p (both jugglers do the same sequence, offset by half a period).

3 3 4p - two self throws (value 3), then a pass (value 4, goes to partner)
Mean: (3+3+4)/3 = 10/3 - not an integer. But wait - this is Prechac notation where passes count at half-value for the throwing juggler…

Causal diagram for the 6-club 3-count passing pattern. Juggler A (top) and Juggler B (bottom) exchange objects every third beat. Passes are shown crossing between rows.

Standard 2-person passing patterns

The named 6-club patterns form a well-understood vocabulary:

Juggling

Three-person patterns: the feed and the W

With three jugglers, the state space expands significantly. The simplest structure is the feed: one juggler (the feeder) passes alternately to each of two feedees, while the feedees only pass to the feeder.

The 3-person feed pattern. The feeder (center) passes to feedee A, then feedee B, alternating. Feedees pass back to feeder only. The feeder's pass rate is double that of each feedee.

The W pattern places three jugglers in a line. The outer two pass to each other across the middle juggler who passes to both. This creates a more complex shared state where the middle juggler must synchronize with two different rhythms simultaneously.

The mathematics of the W pattern cannot be expressed in simple Prechac notation without an extension to three sequences. The International Juggling Association’s pattern library uses a graph-based notation for patterns beyond two jugglers.

The combinatorics of passing

For two jugglers sharing b objects each (total 2b objects), with sequences of period n, the number of valid Prechac patterns grows rapidly with b and n. The exact count requires inclusion-exclusion over the joint state graph of both jugglers.

What the research shows (Beek and Lewbel, 2007; Lewbel’s “Mathematics of Juggling” survey):

For 6 clubs (3 each), period 3, there are exactly 5 essentially distinct 2-person patterns
For 6 clubs, period 6, there are 37 valid patterns
For 8 clubs (4 each), period 3, the count is 14

These are not large numbers - the patterns that exist are countable and have been catalogued. The International Juggling Database (ijdb.com) maintains a searchable index of named passing patterns with their notations and video examples.

Cascade communication: what passing requires cognitively

The cognitive demands of passing are qualitatively different from solo juggling. The juggler must simultaneously:

Maintain their own solo rhythm (the “self” throws)
Predict the partner’s throw timing and trajectory
Synchronize pass release to the partner’s catch window
Adjust in real time when the partner’s timing drifts

Research by Richardson, Marsh, and colleagues on interpersonal coordination (2005-2015) shows that two people maintaining a shared rhythmic task develop coordinative coupling - their movements become synchronized not through explicit communication but through the perceptual-motor feedback loop of watching and responding to each other.

In passing, this coupling is explicit in the prop trajectories: if your partner’s throw is slightly early, the club arrives early, you catch early, and your subsequent throw is early. The error propagates unless actively corrected. Experienced passing pairs learn to make micro-corrections transparently - absorbing the partner’s timing errors without visible disruption to the visual pattern.

“A passing pattern is not two soloists coordinating. It is one shared pattern distributed across two bodies. The objects do not belong to either juggler. They live in the space between, following a schedule that requires both people to hold simultaneously.”