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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 =

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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: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?