### 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?

# All Tangled Up

##### Stage: 3 Challenge Level:

This problem follows on from Twisting and Turning and More Twisting and Turning in which twisting has the effect of adding 1 and turning transforms any number into the negative of its reciprocal.

We can start at 0 and end up at any fraction of the form $$\frac{n}{n+1}$$ by following the sequence: twist, twist, twist, ... , twist, twist, turn, twist

eg. to end up at $\frac{4}{5}$:

twist, twist, twist, twist, twist, turn, twist

to produce:
$0, 1, 2, 3, 4, 5, \frac{-1}{5}, \frac{4}{5}$

Check you can reach $\frac{9}{10}$

The sequence twist, twist, turn, twist, twist, turn, twist, twist, turn, ... , twist, twist, turn, twistwill lead us from 0 to all the fractions of the form $$\frac{1}{n}$$ eg. to end up at $\frac{1}{5}$ (and $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{4}$ along the way):

twist, twist, turn, twist, twist, turn, twist, twist, turn, twist, twist, turn, twist

to produce: 0, 1, 2, $\frac{-1}{2}$, $\frac{1}{2}$, $\frac{3}{2}$, $\frac{-2}{3}$, $\frac{1}{3}$, $\frac{4}{3}$, $\frac{-3}{4}$, $\frac{1}{4}$, $\frac{5}{4}$, $\frac{-4}{5}$, $\frac{1}{5}$

Check you can reach $\frac{1}{10}$
Can you find other sequences of twists and turns that lead to special fractions?

Is it possible to start at 0 and end up at any fraction?