Jannis Ahlers (Long Bay Primary) found 8
transformations:
"The answer is 8. I found this by finding all the possible
positions the shape could end in by only using R, S and there
inverses."
The 8 possible transformations
are:
I, S, S2, S3=S-1, R, R S=S-1R, R S2=S2R, R S3=S R.
There are eight transformations made up only of R, S and their
inverses. Neat way to see this: draw the eight that you think exist, then
note that applying R or S to any of them gives another of them,
so we can't `escape' from these eight. The simplest expressions for the
eight are:
I, S, S2, S3=S-1, R, R S=S-1R, R S2=S2R, R S3=S R.
Notice that R S R-1=S-1. (Of course, R-1=R, so R S R=S-1, and
this can also be written as S R=R S-1.)
So the two expressions simplified are:
S S R S R-1 S R S R-1 = S S(R S R-1)S(R S R-1) = S S S-1S S-1 = S
and
S-1R R S R S R R-1 S R-1 = S-1(R R)S R S(R R-1)S R-1 = S-1 S R S S R-1=(S-1S)R S S R-1=R S(S R)=R S R S-1 = (R S R)S-1=S-1S-1=S-2=S2.