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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, twistwill 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?