Copyright © University of Cambridge. All rights reserved.
In this problem, we shall use four transformations, $I$, $R$, $S$
and $T$. Their effects are shown below.
We write $R^{-1}$ for the transformation that ``undoes'' $R$ (the
inverse of $R$), and $R S$ for "do $R$, then $S$".
We can write $T$ followed by $T$ as $T T$ or $T^2$, and $T$
followed by $T$ followed by $T$ as $T T T$ or $T^3$ and so
on.
Similarly, we can write $S^{-1}S^{-1}$ as $S^{-2}$ and so on.
Try to find simpler ways to write:
$R^2$, $R^3$, $R^4$, $\dots$
$S^2$, $S^3$, $S^4$, $\dots$
$T^2$, $T^3$, $T^4$, $\dots$.
What do you notice?
Can you find a simpler way to write $R^{2006}$ and $S^{2006}$?
Can you describe $T^{2006}$?
Let's think about the order in which we carry out
transformations:
What happens if you do $R S$? Do you think that $S R$ will be the
same? Try it and see.
Is $T^2R$ the same as $R T^2$?
Is $(R T)S$ the same as $S(R T)$?
Try this with some other transformations.
Does changing the order always/sometimes/never produce the same
transformation?
Now let's think about how to undo $R S$. What combination of $I$,
$R$, $S$, $T$ and their inverses might work? Try it and see: does
it work? If not, why not? Can you find a combination of
transformations that does work?
How can you undo transformations like $S T$, $T R$ and $R S^2$?