This problem follows on from Combining Transformations .

We shall concentrate on the transformations $I$, $R$ and $S$ in this question. Here they are again:

We can combine $R$, $S$, $R^{-1}$ and $S^{-1}$ in lots of different ways.

$S S R S R^{-1} S R S R^{-1}$ and $ S^{-1} R R S R S R R^{-1} S R^{-1}$ are two examples of transformations obtained like this.

In fact, there are infinitely many ways to combine them.

How many different transformations can you find made up from combinations of $R$, $S$ and their inverses?

(We shall count two transformations as the same if they have the same effect on all starting shapes.)

Can you be sure that you have found them all?

Convince yourself that $R S = S^3 R$.

In Combining Transformations you found simpler expressions for powers of $R$ and $S$.

Use these and $R S = S^3 R$ to simplify $S S R S R^{-1} S R S R^{-1}$ and $S^{-1}R R S R S R R^{-1} S R^{-1}$.

You might like to look at the article Grouping
Transformations , which explains some of the mathematics behind
these problems.