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Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

All Tangled Up

Age 14 to 18 Challenge Level:

This problem follows on from Twisting and Turning and More Twisting and Turning

In More Twisting and Turning, you were invited to find a strategy to reduce any fraction $\frac ab$ to zero using only the operations Twist ($x \mapsto x+1$) and Turn ($x \mapsto -\frac1x$). In this problem, we will call these operations $T$ for twist, and $R$ for rotate (since 'turn' rather unhelpfully begins with a T too!).

To end up at $\frac{4}{5}$, you can carry out the following sequence of operations: $T,T,T,T,T,R,T$ which could be written more concisely as $T^5RT = \frac45$.

Can you find a sequence of operations that leads to $\frac{9}{10}$?
What about $\frac{23}{24}$?

Can you find a sequence of operations that gets from $0$ to the fraction $\frac{n}{n+1}$?

Now try the following sequences:
  • $T^2RT$
  • $T^2RT^2RT$
  • $T^2RT^2RT^2RT$
What do you notice?
Can you find a way to reach $\frac{1}{10}$?
Can you prove that the pattern will continue?
 
Can you find other patterns that lead to interesting fractions?

Can you prove that it is possible to start at zero and reach any fraction using only the operations $T$ and $R$?