# All tangled up

Can you tangle yourself up and reach any fraction?

*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 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 $R$?**

Here is an interactive tool you can use to try out your ideas. Experiment with the sequences suggested in the problem (remember, $T$ is Twist and $R$ is Turn). What do you notice about the fractions that emerge?

To prove relationships generally, it may be helpful to use algebra.

For the final part of the task, it's useful to work backwards.

Consider the alternative function that maps $x \mapsto x-1$, an inverse twist.

Choose a fraction, and find a way to use the inverse twist and the turn functions to get back to zero.

How does that help you get from zero to your chosen fraction using the original twist and turn functions?

David from St Peter's College in Adelaide, Dakshesk from Richard Huish College and Ruairi from The Skinners' School in the UK all sent in full solutions.

Ruairi explained how to form sequences leading to fractions of the form $\frac{n}{n+1}$:

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Dakshesh's method involved working backwards:

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This is Dakshesk's explanation of how to make fractions of the form $\frac1n$:

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David used this idea to form a proof by induction:

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To prove it is possible to make any fraction, David and Ruairi used the idea of working backwards. Dakshesh contsructed a proof by looking for patterns. All are shown below.

David described a strategy for working backwards:

We can work it backwards, which means finding a method of reducing any fraction to $0$ by using $T^{-1}$, which is $xâ†¦xâˆ’1$, and $R$ (since the inverse of $R$ is also itself)

With any positive fraction, we use operation $T^{-1}$ to reduce that fraction to a negative fraction, and then use $R$ to turn it back to a positive fraction. Once it becomes a positive fraction, then continue use $T^{-1}$ to reduce it to a negative fraction. And once it becomes a negative fraction, use $R$ to turn it back to positive fraction. Notice that, by doing these operations, the denominator will get smaller and smaller until it gets to $1.$

For example, if we start with $\frac{10}7$ ,we operate it by $T^{-2}$ to get $-\frac47$ and use $R$ to get $\frac74$, and then operate it by $T^{-2}$ to get $-\frac14$, then use $R$ to get $4$, and operate it by $T^{-4}$ to get $0.$

(So to get to $\frac{10}7$, repeat the process backwards: $T^4RT^2RT^2$)

Click here to see David's diagram illustrating this method.

Ruairi went into more detail to show that this process leads to zero:

To prove that it is possible to reach any fraction, we only need to prove that a method exists to reach its fraction part

*[e.g. the fraction part of $\frac{10}7 = 1\frac37$ is $\frac37$]*, as we can add necessary [whole numbers] by twisting the required number of times.

Take a fraction $\frac ab, a \lt b$, $a$ and $b$ are both natural numbers.

$\frac ab = 1 - \frac{b-a}b$

*(so applying $T^{-1}$ to $\frac ab$ gives $-\frac{b-a}b$]*

Applying $R^{-1}$

*(to $-\frac{b-a}b$)*gives $\frac{b}{b-a}$

As $b$ and $a$ are positive and $b>a$, $b-a$ is lower than $b,$ so this can be broken up into a [whole number] and fraction part with numerator less than $b$.

By induction if we repeat this, each time the denominator and the numerator decrease until either the denominator becomes $1$ in which case we have an integer which is easily found, or a fraction $\frac1n$, which we have also found a method to find, so it follows that every rational number can have a sequence of the functions $T$ and $R$ produced by applying this method and reading it off in reverse.

A note on efficiency: It is entirely conceivable that for some fractions there are sequences that reach the result in fewer steps, this is just one possible algorithm that could be used.

Dakshesh began by finding some fractions, and then worked from the patterns that emerged. Click here to see Dakshesh's proof.

### Why do this problem?

This problem follows on from Twisting and Turning and More Twisting and Turning and offers students the opportunity to explore sequences of fractions, make generalisations, and prove conjectures. The rope trick in the first problem offers a moment of surprise when the rope becomes untangled, and this prompts curious students to want to explain what is going on. This third problem in the sequence ties up all the loose ends still remaining.

Possible approach

Students will need to be familiar with the operations introduced in Twisting and Turning:**Twist**($x \mapsto x+1$) and

**Turn**($x \mapsto -\frac1x$).

These notes assume that students will have worked on More Twisting and Turning, and developed a general strategy to get to zero, from any fraction. This problem uses T to represent Twist, and R (for rotate) to represent Turn.

**Set the first challenge**:

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 $\frac{n}{n+1}$?

Give students some time to work on this, perhaps using the interactive tool in Getting Started to check their work. Then discuss their strategy and justification for getting fractions of the form $\frac{n}{n+1}$.

**Then set the next challenge**, asking students to try the following sequences:

$T^2RT$

$T^2RT^2RT$

$T^2RT^2RT^2RT$

What do they notice?

Can they find a way to reach $\frac{1}{10}$?

Can they prove that the pattern will continue?

Can they find a way to reach $\frac{1}{10}$?

Can they prove that the pattern will continue?

For students who have met proof by induction, this could be a good opportunity to use that strategy.

The final part of the problem suggests students look for ways to make any fraction, starting from zero. This is quite challenging; the key questions below might provide a useful prompt.

### Key questions

What could you do if you had the function $x \mapsto x-1$, an inverse twist, instead of the original twist function?Can you find a strategy to get from any fraction to zero using an inverse twist and a turn function?

How does that help you get from zero to your chosen fraction using the original twist and turn functions?