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
n
n+1


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

eg. to end up at
4
5

:

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

to produce:

0, 1, 2, 3, 4, 5, -1
5
 , 4
5

Check you can reach
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
1
n

eg. to end up at
1
5

(and
1
2

,
1
3

and
1
4

along the way):

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

to produce:
0, 1, 2,
-1
2

,
1
2

,
3
2

,
-2
3

,
1
3

,
4
3

,
-3
4

,
1
4

,
5
4

,
-4
5

,
1
5

Check you can reach
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?