This problem follows on from Twisting and Turning and More Twisting and Turning in which twisting has the effect of adding 1 and turning transforms any number into the negative of its reciprocal.

We can start at 0 and end up at any fraction of the form

$\frac{n}{n + 1}$

by following the sequence: twist, twist, twist, ... , twist, twist, turn, twist

eg. to end up at

\frac{4}{5}

twist, twist, twist, twist, twist, turn, twist

to produce:
$0, 1, 2, 3, 4, 5, \frac{- 1}{5}, \frac{4}{5}$

Check you can reach $\frac{9}{10}$

The sequence twist, twist, turn, twist, twist, turn, twist, twist, turn, ... , twist, twist, turn, twist

will lead us from 0 to all the fractions of the form

$\frac{1}{n}$

eg. to end up at

\frac{1}{5}

(and

\frac{1}{2}

\frac{1}{3}

and

\frac{1}{4}

along the way):

twist, twist, turn, twist, twist, turn, twist, twist, turn, twist, twist, turn, twist

to produce:
0, 1, 2,

\frac{- 1}{2}

\frac{1}{2}

\frac{3}{2}

\frac{- 2}{3}

\frac{1}{3}

\frac{4}{3}

\frac{- 3}{4}

\frac{1}{4}

\frac{5}{4}

\frac{- 4}{5}

\frac{1}{5}

Check you can reach $\frac{1}{10}$

Can you find other sequences of twists and turns that lead to special fractions?

Is it possible to start at 0 and end up at any fraction?