V. Rolfe sent the following very clear solution to the Beads problem:

1) All three beads are red - RRR
When more beads are added, they will also be red as all the beads are the same colour. The pattern therefore remains the same.

2) Three blue beads - BBB
Three red beads are added as the beads are all the same colour, we then have the scenario described above, with three red beads and the pattern remains red.
Thus BBB becomes RRR

3) Two blue beads and one red bead - BBR (The order is not significant in the circle)
There are two sets of beads of different colours next to each other therefore two blue beads are inserted. The two blue beads lead to the insertion of one red bead.
Thus two blue beads and one red bead are added so the pattern remains constant, although the ring does 'rotate'.

4) Two red beads and one blue bead - RRB
This is the scenario above with two different coloured beads next to each other and one pair of the same colour. Two blue beads and one red bead are therefore inserted. When the original beads are removed the pattern becomes that described above, BBR.

Thank you - you have gone through the possibilities very systematically.