In the cascade, every throw crosses. In 531, every throw still crosses - all three values are odd. 423 breaks that rule for the first time in the typical three-ball progression.
The 4 throw is even. It goes up and comes back to the same hand. The 2 is not thrown at all - it is a hold, a beat where one hand simply retains its ball while the other side of the pattern continues. The 3 is a familiar cascade cross.
The result is a pattern with an asymmetric internal rhythm: one hand throws a high arc to itself, the other hand sits still, then they swap roles and do it again. The pause is not an absence of skill. It is a separate and distinct motor demand.
The mathematics of 423
The Buhler-Graham-Eisenbud-Wright theorem (1994) states that a sequence is a valid juggling pattern if and only if all values (t_k mod n) are distinct, where n is the period length. For 423:
- Period: n = 3
- Values: 4, 2, 3
- Values mod 3: (4 mod 3, 2 mod 3, 3 mod 3) = (1, 2, 0)
- All distinct: yes. Valid.
- Mean: (4+2+3)/3 = 3 balls required.
The average theorem confirms that 423 uses exactly three balls - the same count as the cascade and 531.
The values are not all odd. 4 and 2 are both even. In siteswap with alternating hand throws, an even throw means the ball returns to the same hand that threw it. An odd throw crosses to the other hand. This is why 423 looks and feels different from both the cascade and 531: it contains throws that do not cross, and a beat that is not a throw at all.
The state cycle
In the siteswap state diagram for 3-ball patterns with maximum throw value 4, states are represented as 4-bit binary strings where bit i = 1 means a ball lands i beats from now.
The cascade ground state in 4-bit notation is 1110 - the next three beats each have a ball, and beat four is empty. The 423 cycle visits exactly two distinct states:
Starting from 1110 (ground state), throwing a 4 moves the system to 1101. Throwing a 2 from 1101 returns the system directly to 1110. Throwing a 3 from 1110 keeps the system at 1110 (the ground state is a valid position for the 3 throw, consistent with the cascade).
The two-state cycle 1110 → 1101 → 1110 → 1110 → ... repeats every three beats. Unlike 531, which visits three distinct states, 423 passes through only two: the cascade ground state and one excursion state produced by the high self-throw. The 3-throw beat is a return to familiar ground at every cycle.
This is also why 423 is directly accessible from the cascade without transition throws: the cascade’s ground state is one of the two states in the 423 cycle.
The three throws and their physical character
The 4: At a standard juggling tempo of approximately 0.35 seconds per beat, the 4 throw has a flight time of 4 * 0.35 = 1.4 seconds. Using the parabolic height formula h = g * t² / 8, this gives a peak height of approximately 9.81 * (1.4)² / 8 ≈ 2.4m. In practice, the 4 throw reaches approximately 1.5-2.5m above the hands. It is noticeably higher than a cascade throw but lower than the 5 in 531. Crucially, it returns to the throwing hand: the same hand that releases the ball will catch it 4 beats later.
The 2: A throw of value 2 has a theoretical flight time of 2 * 0.35 = 0.7 seconds. But in practice, a 2 in a two-hand alternating pattern is executed as a hold - the ball is not thrown at all. The hand retains the ball for the duration of that beat and the next, then either throws or holds again. The height is zero. The required action is inaction. This is the first time in the standard juggling progression that a hand is explicitly asked to do nothing for a beat while the pattern continues.
The 3: Identical to a standard cascade throw. Approximately 1.0m peak height at standard tempo. The 3 beat is the rhythm anchor and the only crossing throw in the cycle - the moment where 423 resembles the cascade.
Why the hold is harder than it looks
Schmidt’s schema theory (1975) describes motor control in terms of a generalized motor program parameterized by force, timing, and direction. Every active throw - the 5 in 531, the 4 in 423, the cascade 3 - requires a specific parameterization of this program. The hands generate force; the brain monitors the outcome.
The 2-hold requires something different: the suppression of the throw command. The hand must maintain its grip on the ball, stay in position within the pattern, and produce no ballistic output - all while the rhythm continues. Research on motor timing (Wing and Kristofferson, 1973) demonstrates that maintaining a consistent inter-event interval is an active process, not a passive one. The hand holding a ball in a 2-beat is still participating in the timing structure; it is simply not launching anything.
This is why beginners often turn the 2 into an unintended throw - flipping the ball forward slightly, or releasing it and catching it immediately. The motor system defaults to action. The explicit suppression of that default is itself a learned skill.
“423 is where the juggler learns that not throwing is a skill. The 2-hold requires the hand to remain in the pattern without contributing a throw - a different demand from any active throw, and one that the motor system has to be specifically trained for.”
What 423 introduces structurally
The cascade runs on one throw type, repeated. 531 runs on three different throw types per cycle, all active, all crossing. 423 runs on two active throw types and one non-throw - and one of the active throws is a self-throw that never crosses the centerline.
This makes 423 the first pattern in the standard progression that breaks the “every throw crosses” rule of the cascade. It is also the first pattern where hand activity is explicitly asymmetric per beat: on the hold beat, one hand is doing something (holding) and the other is doing something different (throwing a 3 or setting up for the 4).
The cross at beat 3 (the 3-throw) is the glue that keeps the pattern recognizable. Jugglers describe 423 as feeling like a cascade with one hand repeatedly “pausing” while the other hand throws a high arc. The 3 is where both sides synchronize again before the next self-throw.
Practical technique notes
The most common error in 423 is releasing the 2-hold ball prematurely - accidentally throwing it forward while intending to hold. The fix is deliberate: consciously close the fingers around the ball on the hold beat and wait.
The 4 throw is the second source of errors. Because it does not cross, beginners often aim it incorrectly - throwing it too far across the centerline (turning it effectively into a 3 or 5) or not far enough (dropping it in front of the throwing hand’s column). The 4 should go straight up from the throwing hand and fall back toward the same hand, not toward the opposite side.
Once both are stable, the 3 beat provides the reset: it is exactly a cascade throw and reestablishes the shared rhythm before the next cycle.
Further reading
- Buhler, J., Eisenbud, D., Graham, R., and Wright, C. (1994). “Juggling Drops and Descents.” The American Mathematical Monthly, 101(6), 507-519. The validity theorem that confirms 423 is valid: mod-3 values (1, 2, 0) are all distinct, guaranteeing no two balls land on the same beat.
- Polster, B. (2003). The Mathematics of Juggling. Springer. Chapter 2 covers the state diagram for 3-ball patterns, including the two-state cycle that 423 traces through the ground state and the excursion state produced by the 4-throw.
- Schmidt, R.A. (1975). “A schema theory of discrete motor skill learning.” Psychological Review, 82(4), 225-260. Schema theory explains why the hold (value 2) requires its own learned suppression routine, distinct from the parameterization of active throws.
- Wing, A.M. and Kristofferson, A.B. (1973). “Response delays and the timing of discrete motor responses.” Perception and Psychophysics, 14(1), 5-12. Establishes that maintaining a timing interval is an active cognitive process - directly relevant to the hold beat, which requires the juggler to maintain rhythm without throwing.
On this site: The Mathematics of Siteswap covers the validity theorem and the even/odd rule that explains why the 4 self-throws and the 3 crosses. 531: The First Trick After the Cascade is the other standard first pattern after the cascade - all odd values, all crossing, three distinct heights. The Cascade is the ground state from which 423 is directly accessible.