With 3 beads the only initial states are RRR BBB RBB and BRR

(1) With 3 beads the steady state is RRR.
(2) BBB becomes RRR and then remains RRR for all time
(3) RBB becomes BRB then BBR then RBB again giving a 3-cycle
(4) BRR becomes BRB goes into a 3-cycle as in (3)