Map created from Claude Shannon's juggling theorem for the juggling pattern known as the Cascade. (F+D)H = (V+D)N.
F = time in air
D = time in hand
H = number of hands
V = time hand is vacant
N = number of ballsThus for the three ball Cascade: (3+3)2 = (1+3)3 = 12.

A three ball pattern for one person, we know that, H = 2 and N = 3, thus:
(F+D)2 = (V+D)3.
The LCD of 2 and 3 is 6, and 6*2 = 12. The only option for x*2 = y*3 = 12 is that x = 6 and y = 4.
This means that the sum of F and D must be 6 and the sum of V and D must 4:
(6)2 = (4)3.
The only options for D in (V+D)3 = 12 3, 2, and 1.
If D = 2 then F must equal 4 and V must equal 2. (4+2)2 = (2+2)3 = 12
If D = 1 then F must equal 5 and V must equal 3. (5+1)2 = (3+1)3 = 12
Presumably neither of the above are possible since there are only 3 balls, there can't be values above three.
If D = 3 then F must equal 3 and V must equal 1. (3+3)2 = (1+3)3 = 12      1st 2nd 3rd
hand: D--VD--VD--V
ball: D--F--D--F--
      R     L

(F+D)H=(V+D)N
(3+3)2=(1+3)3
6×2=4×3
12=12
Own work
Hyacinth