Twisting and turning
Problem
Twisting and Turning printable sheet
This problem introduces a trick that can be done with two skipping ropes. It was invented by the mathematician John Conway.
Imagine four people standing at the four corners of a square. Let's call the corners after the compass directions NW, NE, SE and SW. To begin, NE and NW each hold an end of one rope, and SE and SW each hold the end of another rope.
There are two operations that the people can perform, twisting and turning.
In a twist, the person standing at NE swaps places with the person standing at SE, with NE lifting the rope over SE as they swap.
In a turn, everyone moves clockwise one place - NE to SE, SE to SW, SW to NW and NW to NE.
At every stage, the 'tangle' in the ropes can be represented by a number. The initial untangled state is represented by 0.
Twisting has the effect of adding 1: $$x \mapsto x+1$$
Turning transforms any number into the negative of its reciprocal: $$x \mapsto -\frac1x$$
Take a look at this video to see an example of the ropes being tangled and untangled:
If you can't access YouTube, here is a direct link to the video: Twisting and Turning.mp4
If you can't see the video, click below for a description of the process.
The operations are performed in the following sequence, resulting in a new fraction at each stage:
| Operation | Fraction |
|---|---|
| Twist | $1$ |
| Twist | $2$ |
| Turn | $-\frac12$ |
| Twist | $\frac12$ |
| Twist | $1\frac12$, or $\frac32$ |
| Twist | $2\frac12$, or $\frac52$ |
| Turn | $-\frac25$ |
| Twist | $\frac35$ |
| Twist | $1\frac35$, or $\frac85$ |
| Twist | $2\frac35$, or $\frac{13}5$ |
| Turn | $-\frac5{13}$ |
With the ropes now tangled, the team begin the process of untangling them again!
| Operation | Fraction |
|---|---|
| Twist | $\frac8{13}$ |
| Turn | $-\frac{13}8$ or $-1\frac58$ |
| Twist | $-\frac58$ |
| Twist | $\frac38$ |
| Turn | $-\frac83$ or $-2\frac23$ |
| Twist | $-\frac53$ or $-1\frac23$ |
| Twist | $-\frac23$ |
| Twist | $\frac13$ |
| Turn | $-3$ |
| Twist | $-2$ |
| Twist | $-1$ |
| Twist | $0$ |
At the end of the video, once again the ropes are untangled. Hurrah!
Now that you understand how the two operations work, try this:
Starting at zero (with both ropes parallel), what would you end with after following this sequence of moves?
Can you work out a sequence of moves that will take you back to zero?
You may find it easier to call the turning move "Rotate" so that you can abbreviate the operations as T and R.
More Twisting and Turning follows on from this problem.
Getting Started
Twisting has the effect of adding 1: $$x\mapsto x + 1$$ Turning transforms any number into the negative of its reciprocal: $$x\mapsto -\frac{1}{x}$$ Starting at zero, these five moves: Twist, twist, twist, turn, twist
Can you continue from there and then return to zero? You might find it helpful to record each step in a table.
The team use a strategy to help them get back to zero.
Can you figure out how they decide when to stop twisting and start turning?
If you want to have a go at the trick for yourself, but don't have enough people or skipping ropes, you can also perform the tangling and untangling process using a small piece of card and two pieces of string.
Student Solutions
Ben from the UK, Arkadiusz from the Costello School in the UK and Ved from WBGS in the UK worked out how the rope ends up after the series of moves
Having explored the use of twisting and turning I used the example given to
help me solve the first problem. I found an answer of $-\frac8{21}$ having gone through the sequence of
$1,2,3,-\frac13,\frac23,1 \frac23,2 \frac23,-\frac38,\frac58,1 \frac58,2
\frac58$ to finally get to $-\frac8{21}$
Arkadiusz and Ved also worked out how to get back to $0$. This is Arkadiusz' work:
Mohit from Burnt Mill Academy, Harlow in the UK untangled ropes starting from a different state:
TANGLE :
$-\frac{11}{30}$
UNTANGLE:
$-\frac{11}{30}+1=\frac{19}{30}, \\
\frac{19}{30} \rightarrow -\frac1x=-\frac{30}{19},\\
-\frac{30}{19}+1=-\frac{11}{19},\\
-\frac{11}{19}+1=\frac8{19},\\
\frac8{19}\rightarrow -\frac1x=-\frac{19}8, \\
-\frac{19}8+1=-\frac{11}8, \\
-\frac{11}8+1=-\frac38, \\
-\frac38+1=\frac58,\\
\frac58\rightarrow-\frac1x= -\frac85,\\
-\frac85+1=-\frac35, \\
-\frac35+1=\frac25,
\frac25\rightarrow-\frac1x=-\frac52, \\
-\frac52+1=-\frac32,\\
-\frac32+1=-\frac12,\\
-\frac12+1=\frac12, \\
\frac12\rightarrow-\frac1x=-2, \\
-2+1=-1, \\
-1=1=0$
T,R,T,T,R,T,T,T,R,T,T,R,T,T,T,R (T,T)
Arkadiusz described a method to untangle ropes starting from any position:
We keep twisting until we get a positive value, then we turn and repeat over and over until we get zero.
Teachers' Resources
Why do this problem?
This problem introduces an intriguing trick which provides a context for practising manipulation of fractions. Watching the video, or perhaps trying the trick out for themselves, can engage students' curiosity, and lead to some intriguing mathematics to explore and explain.Possible approach
Show the video in the problem, and explain the two functions:Twisting has the effect of adding 1: $$x \mapsto x+1$$
Turning transforms any number into the negative of its reciprocal: $$x \mapsto -\frac1x$$
You may wish to pause the video at each stage and invite students to work out what the next fraction will be. Below is a useful table showing the operation and the fraction at each stage of the video.
| Operation | Fraction |
|---|---|
| Twist | $1$ |
| Twist | $2$ |
| Turn | $-\frac12$ |
| Twist | $\frac12$ |
| Twist | $1\frac12$, or $\frac32$ |
| Twist | $2\frac12$, or $\frac52$ |
| Turn | $-\frac25$ |
| Twist | $\frac35$ |
| Twist | $1\frac35$, or $\frac85$ |
| Twist | $2\frac35$, or $\frac{13}5$ |
| Turn | $-\frac5{13}$ |
If students have grasped the two operations, you could challenge them to work out a sequence of twists and turns to get back from $-\frac5{13}$ to $0$. Alternatively, watch the rest of the video to see the surprising moment when the ropes disentangle. Here is the sequence:
| Operation | Fraction |
|---|---|
| Twist | $\frac8{13}$ |
| Turn | $-\frac{13}8$ or $-1\frac58$ |
| Twist | $-\frac58$ |
| Twist | $\frac38$ |
| Turn | $-\frac83$ or $-2\frac23$ |
| Twist | $-\frac53$ or $-1\frac23$ |
| Twist | $-\frac23$ |
| Twist | $\frac13$ |
| Turn | $-3$ |
| Twist | $-2$ |
| Twist | $-1$ |
| Twist | $0$ |
Now that students understand the two operations, set them the challenge in the problem - working out the fraction for each stage of the sequence, and working out the sequence to get back to zero:
To finish off, students could perform their sequence to tangle and untangle their own set of skipping ropes, before moving on to More Twisting and Turning. Alternatively, they could perform the sequence individually using string and card, as shown below.
Key questions
What fraction do you reach next if you twist?What fraction do you reach next if you turn?
How will you decide whether to twist or turn next?