# Stringing it Out

Explore the transformations and comment on what you find.

Image

The invitation is for you to explore the following transformations and comment on what you find:

### First

T(x) = y

T(y) = xy

This transformation has the following successive effect (starting with x):

x

y

xy

yxy

xyyxy

...

...

### Next

R(x) = xy

R(y) = z

R(z) = xy

This transformation has the following successive effect:

x

xy

xyz

xyzxy

xyzxyxyz

...

...

### Then

S(w) = wx

S(x) = y

S(y) = wz

S(z) = y

This transformation has the following successive effect:

w

wx

wxy

wxywz

...

...

So what do you notice about this pattern and can you explain why it is occurring?

### Now

Can you suggest an appropriate set of transformations for the following elements that would have the same effect?

T(v) = ?

T(w) = ?

T(x) = ?

T(y) = ?

T(z) = ?

$$T(x) = y $$ $$T(y) = xy$$This transformation has the following successive effect (starting with $x$): $$x \Rightarrow T(x) = y$$ $$ y \Rightarrow T(y) = xy$$ $$ xy \Rightarrow T(x)T(y) = yxy$$ $$ yxy \Rightarrow T(y)T(x)T(y) = xyyxy$$ $$ xyyxy \Rightarrow T(x)T(y)T(y)T(x)T(y) = \dots$$

No solutions were offered to this problem this month.