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More Twisting and Turning

Age 11 to 16 Challenge Level:

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...

Choose a fraction to start from. 
From your chosen fraction, can you find a sequence of twists and turns that get you back to zero? Remember, twisting: $$x \mapsto x+1$$ and turning: $$x \mapsto -\frac1x$$ 

Perhaps you might like to start with a negative fraction containing a $2$ as the denominator, such as: $-\frac{5}{2}$ or $-\frac{17}{2}$ or $-\frac{23}{2}$
Can you find a way to get back to zero?

Try to describe an efficient strategy for disentangling any fraction of the form $$-\frac{n}{2}$$

How does this help you get back to zero from a positive fraction with 2 as the numerator, such as $\frac{2}{7}$ 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.