Stringing it Out
Explore the transformations and comment on what you find.
Age
14 to 16
Challenge level
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.