Application: The rotation game

On some cell phones, in particular the Nokia 7160 (tm), there is a game called ``Rotation''. It is a $3\times 3$ grid, of the form

$$\begin{array}{ccc} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{array} .$$

The letters could be scrambled randomly in an actual game. The allowed moves are rotations of the following form, or combinations thereof: In cycle notation,

• $r_1=(a_1,a_2 ,a_5 ,a_4 )$, which sends the above grid to
  $$\begin{array}{ccc} a_4 & a_1& a_3 \\ a_5 & a_2 & a_6 \\ a_7 & a_8 & a_9 \end{array} ,$$

• $r_2=(a_2 ,a_3 ,a_6 ,a_5 )$, which sends the above grid to
  $$\begin{array}{ccc} a_1& a_5 & a_2 \\ a_4 & a_6 & a_3 \\ a_7 & a_8 & a_9 \end{array} ,$$

• $r_3=(a_4 ,a_5 ,a_8 ,a_9 )$, which sends the above grid to
  $$\begin{array}{ccc} a_1& a_2 & a_3 \\ a_7 & a_4 & a_6 \\ a_8 & a_5 & a_9 \end{array} ,$$

• $r_4=(a_5 ,a_6 ,a_9 ,a_8 )$, which sends the above grid to
  $$\begin{array}{ccc} a_1& a_2 & a_3 \\ a_4 & a_8 & a_5 \\ a_7 & a_9 & a_6 \end{array} .$$

In other words, a legal move in the Rotation game is any permutation of the form $r_{i_1}r_{i_2}...r_{i_k}$, where $1\leq i_j\leq 4$ for all $1\leq j\leq N$. Question: Can each permutation of $\{1,2,...,9\}$ be expressed in the form $r_{i_1}r_{i_2}...r_{i_k}$, where $1\leq i_j\leq 4$ for all $1\leq j\leq N$? This question turns out to be to answer using group theory and a CA system such as GAP, MAGMA, or MAPLE. The next chapter provides the necessary background.