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.