§ The Zen of juggling three balls

• Hold one ball in the left hand A, two in the right hand B, C. This initial configuration is denoted [A;;B,C].
• throw B from the right hand to the left hand. This configuration is denoted by [A;B←;C] where the B← is in the middle since it is in-flight, and has ←since that's the direction its travelling.
• When the ball B is close enough to the left hand that it can be caught, throw ball A. Thus the configuration is now [;(A→)(B←);C].
• Now catch ball B, which makes the configuration [B;A→;C].
• With the right hand, throw C (to anticipate catching A). This makes the configuration [B;(A→)(C←);]
• Now catch the ball A, which makes the configuration [B;C←;A].
• See that this is a relabelling of the state right after the initial state. Loop back!

§ The Zen

• The key idea is to think of it as (1) "throw (B)" (2) "throw (A), catch (B)", (3) "throw (C), catch (A)", and so on.
• The cadence starts with a "throw", and then settles into "throw, catch", "throw catch", "throw, catch", ...
• This cadence allows us to actually succeed in the act of juggling. It fuses the hard parts of actually freeing a hand and accurately catching the ball. One can then focus attention on the other side and solve the same problem again.