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This problem is about an iterative process.
To iterate means to repeat, so an iterative process involves
repeating something many times.
Imagine some dough, Plasticine or BluTack, something that can
be made into a strip then stretched.
We are going to take a length of this material, which we'll
regard as the unit length, and put a mark at some fraction distance
along it.
Now we are going to follow a procedure and see where our mark
ends up.

The material is folded in the middle so that the bottom reaches
back to the top.

The material is now only half a unit in length and twice as fat,
so it is rolled out or stretched uniformly to become one unit in
length again.

Finally we'll note the new position of our mark.
And that's the process we'll be repeating.
Now let's try with an actual fraction.
Starting for example at $\frac{1}{5}$
First Iteration : We fold to get
then roll out, back to unit length
The new position is $\frac{2}{5}$
Second Iteration : fold again
and roll back out to unit length
The position is now $\frac{4}{5}$
Third Iteration : fold again
and roll back out to unit length
The position is now $\frac{2}{5}$ for the second time
But we know what happens after $\frac{2}{5}$, it goes to
$\frac{4}{5}$, then $\frac{2}{5}$ again, and so on for ever.
So what are you invited to investigate?

What happens for other start fractions?

Does everything go to a loop?

What size loops appear and for what fractions?
In fact you'll want to describe those loops very carefully.
This problem came to our attention via an ATM workshop led by Dave
Hewitt, from the School of Education, University of Birmingham.
We appreciate his permission to pass it on.