With 2 beads RB changes to BB which changes to RR (the steady state) so this
always reduces to a steady state.
<br />
<br />
With 3 beads the only initial states are RRR BBB RBB and BRR
<br />
<br />
(1) With 3 beads the steady state is RRR 

<br />(2) BBB becomes RRR and then remains RRR for all time 

(3) RBB becomes BRB then BBR then RBB again giving a periodic 3-cycle 

(4) BRR becomes BRB goes into a periodic 3-cycle as in (3)
<br />
<br />
Generalising this note that for any number of beads if all the beads are red
RRR...R this is a steady state. If all the beads are blue then at the first
step they all become red and reach the steady state.
<br />
<br />
To encode this process we can write 0 for the red beads and 1 for the blue
beads and denote the sequence of <i>n</i> beads by the vector (<i>x</i><sub>0</sub>,&#x00A0;<i>x</i><sub>1</sub>,&#x00A0;...<i>x</i><sub><i>n</i></sub>) where all the <i>x</i>s are 0 or 1.
<br />
This vector is mapped to (<i>x</i><sub>0</sub>+<i>x</i><sub>1</sub>, <i>x</i><sub>1</sub>+<i>x</i><sub>2</sub>, ... <i>x</i><sub><i>n</i></sub>+<i>x</i><sub>0</sub>) where addition is
modulo 2. Note that 1 + 1 =0 and 0 + 0 = 0 which corresponds to replacing two
of the same colour by a red.
<br />
Also 1 + 0 = 1 and 0 +1 = 1 which corresponds to replacing two of different colours by a blue.
<br />
<br />
For 4 beads there are 6 positions which all reduce to the steady state with all red beads.
<br />
<br />
(1) 1111 <font face="symbol">®</font
> 0000 (6) 

<br />(2) 0111 <font face="symbol">®</font
> 1001 (3) <font face="symbol">®</font
> (4) <font face="symbol">®</font
> (1) <font face="symbol">®</font
> (6)

<br />(3) 0011 <font face="symbol">®</font
> 0101 (4) <font face="symbol">®</font
> (1) <font face="symbol">®</font
> (6)

<br />(4) 0101 <font face="symbol">®</font
> 1111 (1) <font face="symbol">®</font
> (6)

<br />(5) 0001 <font face="symbol">®</font
> 0011 (3) <font face="symbol">®</font
> (4) <font face="symbol">®</font
> (1) <font face="symbol">®</font
> (6)

<br />(6) 0000 <font face="symbol">®</font
> 0000 (6)
<br />
<br />
For 5 beads there are 8 positions two reducing to the steady state with all red beads and the other six cycling between 3 positions.
<br />
<br />
(1) 11111 <font face="symbol">®</font
> 00000 (8) 

<br />(2) 01111 <font face="symbol">®</font
> 10001 (5)(6)(2) which is a 3-cycle 

<br />(3) 00111 <font face="symbol">®</font
> 01001 (6) then into the 3-cycle (2)(5)(6)

<br />(4) 01011 <font face="symbol">®</font
> 11101 (2) then into the 3-cycle (5)(6)(2) 

<br />(5) 00011 <font face="symbol">®</font
> 00101 (6)(2)(5) again the 3-cycle 

<br />(6) 00101 <font face="symbol">®</font
> 01111 (2)(5)(6) again the 3-cycle 

<br />(7) 00001 <font face="symbol">®</font
> 00011 (5) then into the 3-cycle (6)(2)(5) 

<br />(8) 00000 <font face="symbol">®</font
> 00000 (8) which is the steady state
<br />
<br />
For 6 beads there are 13 positions three reducing to the steady state with
all red beads, three reducing to a steady state with 2 red and 4 blue beads
nd the other seven eventually cycling between 2 positions.
<br />
<br />
(1) 111111 <font face="symbol">®</font
> 000000 (13) 

<br />(2) 011111 <font face="symbol">®</font
> 100001 (9)(10)(3) the 2-cycle 

<br />(3) 001111 <font face="symbol">®</font
> 010001 (10)(3) is a 2-cycle 

<br />(4) 010111 <font face="symbol">®</font
> 111001 (3)(10) the 2-cycle 

<br />(5) 011011 <font face="symbol">®</font
> 101101 (5) which is a steady state 

<br />(6) 000111 <font face="symbol">®</font
> 001001 (11)(5)which is a steady state 

<br />(7) 001011 <font face="symbol">®</font
> 011101 (4)(3)(10)the 2-cycle 

<br />(8) 010101 <font face="symbol">®</font
> 111111 (1)(13) the red steady state 

<br />(9) 000011 <font face="symbol">®</font
> 000101 (10)(3) the 2-cycle 

<br />(10)000101 <font face="symbol">®</font
> 001111 (3)(10) the 2-cycle 

<br />(11)001001 <font face="symbol">®</font
> 011011 (5) which is a steady state 

<br />(12)000001 <font face="symbol">®</font
> 000011 (9)(10)(3) the 2-cycle 

<br />(13)000000 <font face="symbol">®</font
> 000000 (13) which is the steady state with all red beads