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### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

# All Tangled Up

### 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**:

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:

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?

### Possible extension

Further reading about John Conway's rational tangles can be found in the articles Tangles and Symmetric Tangles.

### Possible support

Encourage students to be playful and explore using the interactive tool in Getting Started.

Or search by topic

Age 14 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions
- Teachers' Resources

Possible approach

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.

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

$T^2RT$

$T^2RT^2RT$

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

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.

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?