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## 'More Twisting and Turning' printed from http://nrich.maths.org/

This problem follows on from

Twisting
and Turning in which

twisting has the effect of adding $1$
and

turning transforms any
number into the negative of its reciprocal.

It would be nice to have a
strategy for disentangling any tangled ropes...
I wonder if it is always possible
to disentangle them...
Before reading on, select a few fractions and try to get back to
$0$.

You could consider ropes that have been tangled up and have left
you with a negative fraction containing a $2$ as the
denominator.

e.g: $-\frac{5}{2}$ or $-\frac{17}{2}$ or $-\frac{23}{2}$ How would
you disentangle them?

Try to describe an efficient strategy for disentangling any
fraction of the form $$-\frac{n}{2}$$ Can this help you disentangle
any positive fraction containing a 2 as the numerator?

eg: $\frac{2}{7}$ or $\frac{2}{15}$ or $\frac{2}{32}$

Next, you could consider ropes that have been tangled up and have
left you with a negative fraction containing a $3$ as the
denominator

e.g: $-\frac{5}{3}$ or $-\frac{17}{3}$ or $-\frac{23}{3}$

Try to describe an efficient strategy for disentangling any
fraction of the form $$-\frac{n}{3}$$ and use this to suggest a
strategy for disentangling any fraction of the form $$\frac{3}{n}$$
Next, you could consider ropes that have been tangled up and have
left you with negative fractions containing $4, 5, 6
\ldots$as the denominator, or positive fractions
containing $4, 5, 6 \ldots$ as the numerator.

Can you develop a strategy for
disentangling any tangled ropes, irrespective of the fraction you
have ended up with?
You may want to take a look at

All
Tangled Up after this.