### Rationals Between...

What fractions can you find between the square roots of 65 and 67?

### There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### Tweedle Dum and Tweedle Dee

Two brothers were left some money, amounting to an exact number of pounds, to divide between them. DEE undertook the division. "But your heap is larger than mine!" cried DUM...

# More Twisting and Turning

##### Age 11 to 16Challenge 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.