Question 113
Starting with any $n$-tuple $R_0$, $n \gt 1$, of symbols from $A$, $B$, $C$, we define a sequence $R_0, R_1, R_2, \dots$, according to the following rule: If $R_j = (x_1, x_2, \dots, x_n)$, then $R_{j+1} = (y_1, y_2, \dots, y_n)$, where $y_i = x_i$ if $x_i = x_{i+1}$ (taking $x_{n+1} = x_1$) and $y_i$ is the symbol other than $x_i, x_{i+1}$ if $x_i \neq x_{i+1}$. Find all positive integers $n \gt 1$ for which there exists some integer $m \gt 0$ such that $R_m = R_0$.