### 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...

### Lower Bound

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

### Sum Equals Product

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for any whole numbers?

# More Twisting and Turning

##### Stage: 3 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...

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.